Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 1 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 4 GIỚI HẠN A. L TH T I. DÃY SỐ CÓ GIỚI HẠN 0. 1. Định nghĩa. Ta nói rằng dãy số n u có giới hạn 0 nếu với mọi số dương nh ỏ bao nhiêu tùy ý cho trước, mọi số hạng của dãy số, kể từ số hạng nào đó tr ở đi, đ ều có giá trị tuyệt đ ối nhỏ hơn s ố dương đó. Bằng cách sử dụng các kí hiệu toán học, định nghĩa trên có thể viết như sau: 00 lim 0 0, : nn u n n n u . Kí hiệu: lim 0 n u hoặc lim 0 n u hoặc 0 n u Ví dụ 1. Chứng min h dãy số 1 45 n n u n sau đây có giới hạn là 0. 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Nhận xét lim 0 lim 0. nn uu Nếu n u có 0 n u , * n thì lim lim0 0 n u . Cho hai dãy số n u và n v . Nếu lim 0 nn n uv v thì lim 0 n u . Đây là một nhận xét quan trọng để chứng minh giới hạn bằng 0bằng định nghĩa. (giới hạn kẹp ). 3. Các dãy số có giới hạn 0 thường gặp. 1 lim 0 n 1 lim 0 n 0 lim 0 C n với C là hằng số 1 lim 0 k n n k 1 lim 0 k n 2, kk lim 0 n q 1. q 4. Ví dụ minh họa. Ví dụ 2 . Chứng minh rằng các dãy số với số hạng tổng quát sau đây có giới hạn 0 a). 1 32 n n u n . b). 3 sin 2 2 n nn u n . c). 1 cos n n n u n . d.) 2 3sin 4cos 21 n nn u n . 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BÀI 1. GIỚI HẠN CỦA DÃY SỐ Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 2 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... II. DÃY SỐ CÓ GIỚI HẠN HỮU HẠN. 1. Định nghĩa. Ta nói dãy số n u có giới hạn là số thực L nếu lim 0 n uL . Khi đó ta vi ết lim n n uL , viết tắt là lim n uL hoặc lim n uL . Nhận xét: Để chứng minh dãy số n u có giới hạn là số thực L ta chuyển về việc đi ch ứng minh lim 0 n uL . lim nn u a u a nhỏ bao nhiêu cũng được với n đủ lớn. Không phải mọi dãy số đều có giới hạn hữu hạn. Ví dụ 2. Chứng minh rằng a). 3 3 lim 1 1 n n . b). 2 2 3 2 1 lim 22 nn nn . 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Ví dụ 3. Chứng minh rằng a). 3.3 sin3 lim 3 3 n n n . b). 2 1 lim 2 n n n . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 3 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Một số định lý. Định lý 1. ( tìm giới hạn của hàm trị tuyệt đ ối hoặc căn th ức) Giả sử lim n uL . Khi đó lim n uL và 3 3 lim n uL . Nếu 0 n u với mọi n thì 0 L và lim n uL . Định lý 2. Giả sử lim n uL , lim n vM và C là một hằng số. Khi đó lim nn u v L M . lim . . nn u v LM lim n Cu CL . lim n n u L vM với 0 M . limcc (c là hằng số). Nhận xét. Cho ba dãy số , nn uv và n w . Nếu , n n n u v w n và lim lim , nn u w a a thì lim n va (gọi định lí kẹp). Điều kiện để một dãy số tăng hoặc dãy số giảm có giới hạn hữu hạn: Một dãy số tăng và bị chặn trên thì có giới hạn hữu hạn. Một dãy số giảm và bị chặn dưới thì có giới hạn hữu hạn. 3. Tổng của cấp số nhân lùi vô hạn Cho cấp số nhân n u có công bội q và thỏa 1 q . Khi đó tổng 1 2 3 n S u u u u được gọi là tổng vô hạn của cấp số nhân và 1 1 1 lim lim 11 n n uq u SS qq . Vậy cấp số nhân n u có công bội q thỏa mãn 1 q thì 1 12 ... 1 u S u u q . Ví dụ 4. Tính các tổng sau a). 2 1 1 1 ... ... 3 3 3 n S b). 1 1 1 1 1 . 2 4 2 n n S c). 16 8 4 2 ... S Lời giải Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 4 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Ví dụ 5 . Hãy biểu diễn các số thập phân vô hạn tuần hoàn sau dưới dạng phân số a). 0,353535... A . b). 5,231231... B . 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III. DÃY SỐ CÓ GIỚI HẠN VÔ CỰC 1. Dãy số có giới hạn Định nghĩa. Ta nói rằng dãy số n u có giới hạn là nếu với mỗi số dương tùy ý cho trư ớc, mọi số hạng của dãy số, kể từ số hạng nào đó tr ở đi, đều lớn hơn s ố dương đó. Khi đó ta vi ết lim n u hoặc lim n u . Từ định nghĩa, ta có các k ết quả 3 lim ; lim ; lim n n n . 2. Dãy số có giới hạn Định nghĩa. Ta nói rằng dãy số n u có giới hạn là nếu với mỗi số âm tùy ý cho trư ớc, mọi số hạng của dãy số, kể từ số hạng nào đó tr ở đi, đều nhỏ hơn s ố âm đó. Khi đó ta vi ết lim n u hoặc lim n u . Nhận xét. Nếu lim n u thì lim n u . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 5 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Các dãy số có giới hạn hoặc được gọi chung là các dãy số có giới hạn vô cực hay dần đến vô cực. Nếu lim n u thì 1 lim 0 n u . 3. Các quy tắc tìm giới hạn vô cực • Quy tắc nhân: lim n u lim n v lim . nn uv lim n u lim 0 n vL lim . nn uv • Quy tắc chia lim 0 n uL có dấu lim 0, 0 nn vv có dấu lim n n u v 4. Các ví dụ minh họa. Ví dụ 6 . Tìm giới hạn của dãy n u biết: a). 32 3 2 2 n u n n b). 43 23 n u n n n c). 43 2 4 2 1 1 n nn u n d). 1 4 2. 3 n n n u e). 2 2 cos 4 10 n n u n n f). 4 2 2 3 4 1 2 32 n n n n u n n n . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 6 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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B. PHÂN DẠNG VÀ BÀI TẬP MINH HỌA. Dạng 1. Chứng minh dãy số có giới hạn là 0. 1. Phương pháp. Cách 1: Áp dụng đ ịnh nghĩa. Cách 2: Sử dụng các đ ịnh lí sau: Nếu k là số thực dương thì 1 lim 0 k n . Với hai dãy số n u và n v , nếu nn uv với mọi n và lim 0 n v thì lim 0 n u . Nếu 1 q thì lim 0 n q . 2. Bài tập minh họa. Bài tập 1. Chứng minh các dãy số n u sau đây có giới hạn là 0. a). cos 4 3 n n u n b). 3 1 cos 23 n n u n c). 11 1 1 23 n n nn u Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 7 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Bài tập 2. Chứng minh rằng các dãy số với số hạng tổng quát sau đây có giới hạn 0 a). 33 21 n u n n . b). 3 sin 2 4 2 4.5 nn n nn n u . c). 2 sin 2 n nn nn u . d). cos 5 n n n u n n n . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 8 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Bài tập 3. Cho dãy số n u với 3 n n n u . a). Chứng minh rằng 1 2 3 n n u u với mọi * n . b). Bằng phương pháp quy nạp chứng minh rằng 2 0 3 n n u với mọi * n . c). Dãy n u có giới hạn 0. 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Câu hỏi trắc nghiệm . Câu 1. Kết quả của giới hạn sin 5 lim 2 3 n n bằng: A. 2. B. 3. C. 0. D. 5 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 9 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Câu 2. Có bao nhiêu số tự nhiên chẵn k để 1 2 cos 1 lim . 22 k nn n n A. 0. B. 1. C. 4. D. Vô số. 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Câu 3. Kết quả của giới hạn 3sin 4cos lim 1 nn n bằng: A. 1. B. 0 . C. 2 . D. 3 . 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Câu 4. Kết quả của giới hạn 2 cos 2 lim 5 1 nn n bằng: A. 4 . B. 1 . 4 C. 5 . D. 4. 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Câu 5. Kết quả của giới hạn 23 lim sin 2 5 n nn là: A. . B. 2. C. 0 . D. . 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Câu 6. Giá trị của giới hạn 1 lim 4 1 n n bằng: A. 1. B. 3 . C. 4 . D. 2 . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 10 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 7. Cho hai dãy số n u và n v có 2 1 1 n n u n và 2 1 . 2 n v n Khi đó lim nn uv có giá trị bằng: A. 3 . B. 0 . C. 2 . D. 1. 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Dạng 2. Dùng định nghĩa chứng minh dãy số n u có giới hạn hữu hạn L . 1. Phương pháp. Ta biến đ ổi lim n uL về dạng tìm lim có giới hạn bằng 0 tức là chứng minh lim 0 n uL . Kết luận lim n uL . 2. Bài tập minh họa. Bài tập 4. Chứng minh: a). 2 3 1 lim 4 5 2 n n b). 4.3 5.2 2 lim 6.3 3.2 3 nn nn c). 2 lim 2 1 n n n . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 11 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Dạng 3. Tìm giới hạn của dãy n u có giới hạn hữu hạn bằng quy tắc, định lý. Bài toán 1. Dãy n u là một phân thức hữu tỉ dạng n Pn u Qn (với , P n Q n là hai đa thức). 1. Phương pháp: Chia cả tử và mẫu cho k n với k n là lũy thừa có số mũ lớn nhất của Pn và Qn (hoặc rút k n là lũy thừa có số mũ lớn nhất của Pn và Qn ra làm nhân tử) sau đó áp dụng các định lý về giới hạn. Nếu k là số thực dương thì 1 lim 0 k n . Nếu 1 q thì lim 0 n q . 2. Bài tập minh họa. Bài tập 5 . Tìm giới hạn của dãy n u biết: a). 2 2 2 3 1 53 n nn u n b). 32 43 2 3 4 4 n nn u n n n c). 42 2 23 2 1 1 3 2 1 n n n n u n n n d). 22 11 2 2 3 n u n n n e). 2 3 32 2 1 3 4 4 2 2 n nn u nn f). 2 21 23 n nn u nn Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 12 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 13 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 6 . Tìm các giới hạn sau a). 2 2 42 lim 21 nn nn . b). 2 22 31 lim 2 1 2 3 1 n n n n n . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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.................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 4. Câu hỏi trắc nghiệm . Câu 8. Giá trị của giới hạn 2 3 lim 4 2 1 nn là: A. 3 . 4 B. . C. 0 . D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 9. Giá trị của giới hạn 2 3 2 lim 31 nn nn bằng: A. 2 . B. 1. C. 2 . 3 D. 0 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 14 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Câu 10. Giá trị của giới hạn 3 4 3 2 1 lim 4 2 1 nn nn là: A. . B. 0. C. 2 . 7 D. 3 . 4 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 11. Giá trị của giới hạn 1 lim 2 nn n bằng: A. 3 . 2 B. 2. C. 1. D. 0. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 12. Cho hai dãy số n u và n v có 1 1 n u n và 2 . 2 n v n Khi đó lim n n v u có giá trị bằng: A. 1. B. 2. C. 0. D. 3. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 13. Cho dãy số n u với 4 53 n an u n trong đó a là tham số thực. Để dãy số n u có giới hạn bằng 2, giá trị của a là: A. 10. a B. 8. a C. 6. a D. 4. a Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 14. Cho dãy số n u với 2 53 n nb u n trong đó b là tham số thực. Để dãy số n u có giới hạn hữu hạn, giá trị của b là: A. b là một số thực tùy ý. B. 2. b C. không tồn tại . b D. 5. b Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 15 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 15. Tính giới hạn 2 2 5 lim . 21 nn L n A. 3 . 2 L B. 1 . 2 L C. 2. L D. 1. L Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 16. Cho dãy số n u với 2 2 42 . 5 n nn u an Để dãy số đã cho có giới hạn bằng 2, giá trị của a là: A. 4. a B. 4. a C. 3. a D. 2. a Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 17. Tính giới hạn 23 3 3 lim . 2 5 2 nn L nn A. 3 . 2 L B. 1 . 5 L C. 1 . 2 L D. 0. L Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 18. Tìm tất cả các giá trị của tham số a để 24 4 53 lim 0. 1 2 1 n an L a n n A. 0; 1. a a B. 0 1. a C. 0; 1. aa D. 0 1. a Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 19. Tính giới hạn 32 4 2 3 1 lim . 2 1 7 n n n L nn Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 16 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 A. 3 . 2 L B. 1. L C. 3. L D. . L Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 20. Tính giới hạn 23 42 2 2 1 4 5 lim . 3 1 3 7 n n n n L n n n A. 0. L B. 1. L C. 8 . 3 L D. . L Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 21. Tính giới hạn 3 3 1 lim . 8 n L n A. 1 . 2 L B. 1. L C. 1 . 8 L D. . L Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài Toán 2. Dãy n u là một phân thức dạng n Pn u Qn (với , P n Q n là các biểu thức chứa căn của n ). 1. Phương pháp. Bước 1. Chia cả tử và mẫu cho k n với k n là lũy thừa có số mũ lớn nhất của Pn và Qn (hoặc rút k n là lũy thừa có số mũ lớn nhất của Pn và Qn ra làm nhân tử) sau đó áp dụng các định lý về giới hạn. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 17 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Bước 2. Khi đó ta có các k ết quả sau Nếu lim lim n Pn uC Qn giới hạn của dãy số là . C Nếu 0 lim lim n Pn u Qn thì ta nói giới hạn đó có d ạng vô đ ịnh. Khi đó đ ể tính tiếp giới hạn ta phải khử dạng vô đ ịnh bằng các kỹ thuật sau: Đối với căn th ức: nhân lư ợng liên hợp của bậc hai và bậc ba( thêm đuôi) 2 a b a b ab ab a b a b 2 a b a b ab ab a b a b 2 2 3 3 3 3 3 22 22 3 3 3 3 . .. a b a ab b ab ab a ab b a ab b . 2 2 3 3 3 3 3 22 22 3 3 3 3 . .. a b a ab b ab ab a ab b a ab b 2 2 3 3 3 3 3 22 22 3 3 3 3 . .. a b a a b b ab ab a a b b a a b b 2 2 3 3 3 3 3 22 22 3 3 3 3 . .. a b a a b b ab ab a a b b a a b b 22 3 3 3 3 3 3 33 2 2 2 2 3 3 3 3 3 3 3 3 . .. a b a a b b ab ab a a b b a a b b . 22 3 3 3 3 3 3 33 2 2 2 2 3 3 3 3 3 3 3 3 . .. a b a a b b ab ab a a b b a a b b Đối với hàm đa th ức: ta đ ặt nhân tử chung bằng phương pháp phân tích đa thức thành nhân tử. Thừa số chung. Hằng đ ẳng thức đáng nh ớ. . Nhóm, tách và thêm bớt hạng tử. Nhớ: Nếu 2 ax bx c có hai nghiệm 12 , xx thì 2 12 ax bx c a x x x x Nếu bậc ba 32 ax bx cx d thì ta đoán nghiêm 0 xx rồi chia đa th ức đ ể đưa v ề 32 0 ax bx cx d x x f x , với fx là hàm bậc hai. 2. Bài tập minh họa. Bài tập 7 . Tìm giới hạn của dãy n u biết: Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 18 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 a). 2 2 41 93 n n n n u nn b). 2 1 3 45 n nn u n c). 3 2 3 2 24 4 4 1 8 2 3 16 4 1 n n n n u n n n d). 3 23 4 4 3 16 1 n n n n n u n Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 8 . Tìm giới hạn của dãy n u biết: a). 2 35 n u n n n b). 2 9 3 4 3 2 n u n n n c). 332 3 n u n n n d). 3 32 8 4 2 2 3 n u n n n e). 3 2 3 2 4 3 7 8 5 1 n u n n n n f). 3 4 2 6 11 n u n n n Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 19 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 20 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 9 . Tìm các giới hạn sau: a). 2 2 lim 4 3 2 n n n n n n b). 2 3 23 24 4 n n n n u n n n c). 22 lim 2 9 2 n n n n n d). 3 2 2 3 2 lim 2 2 8 3 n n n n n n Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 21 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 22 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 10 . Tìm các giới hạn sau a). 2 9 2 3 lim 43 n n n n . b). 5 4 5 4 3 4 2 lim 23 nn nn . c). 2 lim 4 2 2 n n n . d). 3 3 lim 2 1 . n n n e). 326 42 1 lim 1 nn nn f). 3 2 2 3 lim 2 3 n n n n Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 23 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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.................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 5. Câu hỏi trắc nghiệm . Câu 22. Kết quả của giới hạn 2 91 lim 42 nn n bằng: A. 2 . 3 B. 3 . 4 C. 0. D. 3. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 23. Kết quả của giới hạn 2 4 21 lim 32 nn n bằng: A. 2 . 3 B. 1 . 2 C. 3 . 3 D. 1 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 24 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 24. Kết quả của giới hạn 23 lim 25 n n là: A. 5 . 2 B. 5 . 7 C. . D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 25. Kết quả của giới hạn 14 lim 1 n nn bằng: A. 1. B. 0. C. 1. D. 1 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 26. Biết rằng 2 2 1 lim sin . 4 2 nn ab nn Tính 33 . S a b A. 1. S B. 8. S C. 0. S D. 1. S Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 27. Kết quả của giới hạn 42 10 lim 1 nn là: A. . B. 10. C. 0. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 28. Kết quả của giới hạn 42 22 lim 1 1 n n nn là: A. . B. 1. C. 0. D. . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 25 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 29. Biết rằng 3 32 2 57 lim 3 32 an n bc nn với ,, abc là các tham số. Tính giá trị của biểu thức 3 . ac P b A. 3. P B. 1 . 3 P C. 2. P D. 1 . 2 P Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 30. Kết quả của giới hạn 5 52 lim 200 3 2 nn là: A. . B. 1. C. 0. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 31. Giá trị của giới hạn lim 5 1 nn bằng: A. 0. B. 1. C. 3. D. 5. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 32. Giá trị của giới hạn 2 lim 1 n n n là: A. 1 . 2 B. 0. C. 1. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 26 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 33. Giá trị của giới hạn 22 lim 1 3 2 nn là: A. 2. B. 0. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 34. Giá trị của giới hạn 22 lim 2 2 n n n n là: A. 1. B. 2. C. 4. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 35. Có bao nhiêu giá trị của a để 2 2 2 lim 2 1 0. n a n n a n A. 0. B. 2. C. 1. D. 3. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 36. Giá trị của giới hạn 22 lim 2 1 2 3 2 n n n n là: A. 0. B. 2 . 2 C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 27 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 37. Giá trị của giới hạn 22 lim 2 1 2 n n n n là: A. 1. B. 1 2. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 38. Có bao nhiêu giá trị nguyên của a thỏa 22 lim 8 0 n n n a . A. 0. B. 2. C. 1. D. Vô số. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 39. Giá trị của giới hạn 2 lim 2 3 n n n là: A. 1. B. 0. C. 1. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 28 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Câu 40. Cho dãy số n u với 22 51 n u n an n , trong đó a là tham số thực. Tìm a để lim 1. n u A. 3. B. 2. C. 2. D. 3. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 41. Giá trị của giới hạn 33 33 lim 1 2 nn bằng: A. 3. B. 2. C. 0. D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 42. Giá trị của giới hạn 3 23 lim n n n là: A. 1 . 3 B. . C. 0. D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 43. Giá trị của giới hạn 332 lim 2 n n n bằng: A. 1 . 3 B. 2 . 3 C. 0. D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 29 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 Câu 44. Giá trị của giới hạn lim 1 1 n n n là: A. 1. B. . C. 0. D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 45. Giá trị của giới hạn lim 1 n n n bằng: A. 0. B. 1 . 2 C. 1 . 3 D. 1 . 4 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 46. Giá trị của giới hạn 22 lim 1 3 n n n bằng: A. 1. B. 2. C. 4. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 47. Giá trị của giới hạn 22 lim 1 6 n n n n n là: A. 7 1. B. 3. C. 7 . 2 D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 1. Giới Hạn Dãy số 30 Lớp Toán Thầy –Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 48. Giá trị của giới hạn 2 1 lim 24 nn là: A. 1. B. 0. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 49. Giá trị của giới hạn 2 92 lim 32 n n n n là: A. 1. B. 0. C. 3. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 50. Giá trị của giới hạn 3 3 1 lim 1 nn là: A. 2. B. 0. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 31 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 Bài Toán 3. Dãy n u là một phân thức hữu tỉ dạng n Pn u Qn ( trong đó , P n Q n là các biểu thức chứa hàm mũ ,, n n n abc ,… 1. Phương pháp. Chia cả tử và mẫu cho n a với a là cơ số lớn nhất . Rồi áp dụng kết quả của giới hạn lim 0 n q 1. q 2. Bài tập minh họa. Bài tập 11. Tìm giới hạn của dãy n u biết: a). 24 43 nn n nn u b). 3.2 5 5.4 6.5 nn n nn u c). 21 13 46 5 2.6 nn n nn u d). 2 2 21 32 n n n u e). 1 3 4.5 2.4 3.5 n n n nn u f). 2 1 2 1 2 3 4.5 2 3 5 n n n n n n n u g). 11 2 3 4 lim 2 3 4 n n n n n n h). 2 2 1 2 2 ... 2 lim 1 3 3 ... 3 n n Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 32 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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.................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 3. Câu hỏi trắc nghiệm. Câu 51. Kết quả của giới hạn 2 25 lim 3 2.5 n nn bằng: A. 25 . 2 B. 5 . 2 C. 1. D. 5 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 52. Kết quả của giới hạn 1 1 3 2.5 lim 25 nn nn bằng: A. 15. B. 10. C. 10. D. 15. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 33 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 Câu 53. Kết quả của giới hạn 1 3 4.2 3 lim 3.2 4 nn nn là: A. 0. B. 1. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 54. Kết quả của giới hạn 31 lim 2 2.3 1 n nn bằng: A. 1. B. 1 . 2 C. 1 . 2 D. 3 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 55. Biết rằng 1 2 1 2 5 2 1 2 3 5 lim 1 5.2 5 3 n n n n na c nb với ,, . abc Tính giá trị của biểu thức 2 2 2 . S a b c A. 26. S B. 30. S C. 21. S D. 31. S Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 56. Kết quả của giới hạn 2 22 32 lim 3 3 2 n n n n n n là: A. 1. B. 1 . 3 C. . D. 1 . 4 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 34 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 57. Kết quả của giới hạn lim 3 5 n n là: A. 3. B. 5. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 58. Kết quả của giới hạn 41 lim 3 .2 5.3 nn là: A. 2 . 3 B. 1. C. . D. 1 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 59. Kết quả của giới hạn 1 3 4.2 3 lim 3.2 4 nn n n là: A. 0. B. 1. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 60. Kết quả của giới hạn 1 2 2 3 10 lim 32 n n nn là: A. . B. 2 . 3 C. 3 . 2 D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 35 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 Câu 61. Tìm tất cả giá trị nguyên của a thuộc 0;2018 để 1 4 . 42 l 4 im 2 3 1 0 4 1 nn n n a A. 2007. B. 2008. C. 2017. D. 2016. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 62. Kết quả của giới hạn 2 1 2 lim 3 1 3 n n nn n bằng: A. 2 . 3 B. 1. C. 1 . 3 D. 1 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 63. Kết quả của giới hạn 3 1 cos3 lim 1 n nn n bằng: A. 3 . 2 B. 3. C. 5. D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 64. Có bao nhiêu giá trị nguyên của a thuộc 0;20 sao cho 2 2 11 lim 3 32 n an n là một số nguyên. A. 1. B. 3. C. 2. D. 4. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 36 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 65. Kết quả của giới hạn lim 2.3 2 n n là: A. 0. B. 2. C. 3. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Dạng 4. Tính giới hạn mà dãy n u cho dưới dạng công thức truy hồi. 1. Phương pháp. Đưa dãy số n u về dạng tổng quát rồi làm giống như ba dạng trên. Từ dãy cho dưới dạng truy hồi ta công thức tuy hồi ta đưa về công thức tổng quát. Chứng minh dãy số có giới hạn hữu hạn (có nghĩa chứng minh dãy số tăng và bị chặn trên hoặc dãy số giảm và bị chặn dưới) sau đó dựa vào hệ thức truy hồi để tìm giới hạn. 2. Bài tập minh họa. Bài tập 12. Tìm giới hạn của dãy n u biết: a). 1 1 1 1.2 2.3 ( 1) n u nn b). 2 2 2 2 1 2 3 ( 1)( 2) n n u n n n c). 1 1 1 1 1.4 4.7 7.10 (3 2)(3 1) n u nn d). 2 1 3 5 (2 1) 34 n n u n e). 2 2 2 1 1 1 1 1 ... 1 23 n u n f). 3 3 3 43 12 lim 41 n nn g). 1 1 1 lim 1.2.3 2.3.4 ( 1)( 2) n n n h). 2 2 2 2 4 2.1 3.2 ( 1) ( 1) lim n n n n n k). 1 1 1 lim 1.3 3.5 2 1 2 1 nn Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 37 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 38 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 14. Cho dãy số n u xác định bởi 01 21 1; 6 3 2 0, n n n uu u u u n . Tìm lim 3.2 n n u . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 15. Cho dãy số n u được xác định như sau: 12 21 1, 3 2 1, * n n n uu u u u n . Tính 2 lim n u n . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 39 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 4. Câu hỏi trắc nghiệm. Câu 66. Giá trị của giới hạn 2 13 1 ... 2 2 2 lim 1 n n bằng: A. 1 . 8 B. 1. C. 1 . 2 D. 1 . 4 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 67. Giá trị của giới hạn 2 2 2 1 2 1 lim ... n n n n bằng: A. 0 . B. 1 . 3 C. 1 . 2 D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 68. Giá trị của giới hạn 2 1 3 5 2 1 lim 34 n n bằng: A. 0. B. 1 . 3 C. 2 . 3 D. 1. Lời giải. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 40 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 69. Giá trị của giới hạn 1 1 1 lim ... 1.2 2.3 1 nn là: A. 1 . 2 B. 1. C. 0. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 70. Giá trị của giới hạn 1 1 1 lim ... 1.3 3.5 2 1 2 1 nn bằng: A. 1 . 2 B. 1 . 4 C. 1. D. 2 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 71. Giá trị của giới hạn 1 1 1 lim ...... 1.4 2.5 3 nn bằng: A. 11 . 18 B. 2 . C. 1. D. 3 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 41 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 72. Giá trị của giới hạn 2 2 2 2 1 2 ... lim 1 n nn bằng: A. 4. B. 1. C. 1 . 2 D. 1 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 73. Cho dãy số có giới hạn n u xác định bởi 1 1 2 . 1 , 1 2 n n n u un u Tính lim . n u A. lim 1. n u B. lim 0. n u C. 1 lim . 2 n u D. lim 1. n u Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 74. Cho dãy số có giới hạn n u xác định bởi 1 1 2 . 1 , 1 2 n n u u un Tính lim . n u A. lim 1. n u B. lim 0. n u C. lim 2. n u D. lim . n u Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Dạng 5. Tính giới hạn dựa vào định lý kẹp. 1. Phương pháp. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 42 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Dựa vào định lí: Cho ba dãy số , nn uv và n w . Nếu , n n n u v w n và lim lim , nn u w a a thì lim n va . 2. Bài tập minh họa. Bài tập 16. Tìm giới hạn của dãy n u biết: a). 1 3 5 2 1 2 4 6 2 n n u n b). 2 2 2 1 1 1 12 n u n n n n c). * 1.3.5.7.... 2 1 , 2.4.6...2n n n un d). 1 2 3 ... n n u nn e). 2 2 2 1 1 1 l 4 1 4 2 4 n u n n n n Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 43 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 44 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Dạng 6. Giới hạn có kết quả là vô cực ( hoặc ) 1. Phương pháp. Chia cả tử và mẫu cho k n với k n là lũy thừa có số mũ lớn nhất của Pn và Qn (hoặc rút k n là lũy thừa có số mũ lớn nhất của Pn và Qn ra làm nhân tử) sau đó áp dụng các quy tắc nhân và quy tắc chia). Nếu gặp dạng vô định thì ta khử nhé. Nếu gặp dạng 0 L , là bậc của tử lớn hơn bậc của mẫu thì ta tách cùng bậc đưa về tích. Quy tắc nhân: lim n u lim n v lim . nn uv lim n u lim 0 n vL lim . nn uv Quy tắc chia lim 0 n uL có dấu lim 0, 0 nn vv có dấu lim n n u v 2. Bài tập minh họa. Bài tập 17. Tìm các giới hạn sau a). 5 4 3 32 2 lim 4 6 9 n n n n nn . b). 363 7 5 8 lim 12 n n n n c). lim 2 3 1 nn d). 2 42 1 2 3 lim 1 nn nn e). 4 lim 1 1 nn . f). 3 1 lim 2sin 2 3 3 nn . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 45 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 46 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 18. Tìm các giới hạn sau a). 1 lim 5 3 nn . b). 1 1 36 lim 35 n n n n . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 47 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 5. Câu hỏi trắc nghiệm. Câu 75. Kết quả của giới hạn 5 52 lim 200 3 2 nn là: A. . B. 1. C. 0 . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 76. Giá trị của giới hạn lim 5 1 nn bằng: A. 0. B. 1. C. 3. D. 5. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 77. Giá trị của giới hạn 22 lim 1 3 2 nn là: A. 2. B. 0. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 78. Giá trị của giới hạn 2 1 lim 24 nn là: A. 1. B. 0. C. . D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 48 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 79. Kết quả của giới hạn 3 2 2 lim 13 nn n là: A. 1 . 3 B. . C. . D. 2 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 80. Kết quả của giới hạn 3 2 23 lim 4 2 1 nn nn là: A. 3 . 4 B. . C. 0 D. 5 . 7 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 81. Kết quả của giới hạn 4 3 lim 45 nn n là: A. 0. B. . C. . D. 3 . 4 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 82. Trong các giới hạn sau đây, giới hạn nào bằng 0? A. 3 2 32 lim . 21 n n B. 2 3 23 lim . 24 n n C. 3 2 23 lim . 21 nn n D. 24 42 23 lim . 2 nn nn Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 49 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 83. Dãy số nào sau đây có giới hạn bằng 1 3 ? A. 2 2 2 . 35 n nn u n B. 43 32 21 . 3 2 1 n nn u nn C. 23 32 3 . 91 n nn u nn D. 2 3 25 . 3 4 2 n nn u nn Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 84. Dãy số nào sau đây có giới hạn là ? A. 2 1 . 55 n n u n B. 2 3 2 . 55 n n u nn C. 2 2 2 . 55 n nn u nn D. 2 12 . 55 n nn Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 85. Dãy số nào sau đây có giới hạn là ? A. 2 12 . 55 n nn B. 3 3 21 . 2 n nn u nn C. 24 23 23 . 2 n nn u nn D. 2 2 . 51 n nn u n Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 86. Tính giới hạn 2 lim 3 5 3 . L n n A. 3. L B. . L C. 5. L D. . L Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 50 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Câu 87. Có bao nhiêu giá trị nguyên của tham số a thuộc khoảng 10;10 để 23 lim 5 3 2 L n a n . A. 19. B. 3. C. 5. D. 10. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 88. Tính giới hạn 42 lim 3 4 1 . n n n A. 7. L B. . L C. 3. L D. . L Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 89. Cho dãy số n u với 2 2 2 ... 2 . n n u Mệnh đề nào sau đây đúng ? A. lim . n u B. 2 lim . 12 n u C. lim . n u D. Không tồn tại lim . n u Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 90. Tổng của một cấp số nhân lùi vô hạn bằng 2 , tổng của ba số hạng đầu tiên của cấp số nhân bằng 9 4 . Số hạng đầu 1 u của cấp số nhân đó là: A. 1 3. u B. 1 4. u C. 1 9 . 2 u D. 1 5. u Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 51 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 91. Tính tổng 3 1 1 1 9 3 1 3 9 3 n S . A. 27 . 2 S B. 14. S C. 16. S D. 15. S Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 92. Tính tổng 1 1 1 1 21 2 4 8 2 n S . A. 2 1. S B. 2. S C. 2 2. S D. 1 . 2 S Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 93. Tính tổng 2 4 2 1 3 9 3 n n S . A. 3. S B. 4. S C. 5. S D. 6. S Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 94. Tổng của cấp số nhân vô hạn 1 1 1 1 1 1 , , ,..., ,... 2 6 18 2.3 n n bằng: A. 3 . 4 B. 8 . 3 C. 2 . 3 D. 3 . 8 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 52 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Câu 95. Tính tổng 1 1 1 1 1 1 ... ... 2 3 4 9 2 3 nn S . A. 1. B. 2 . 3 C. 3 . 4 D. 1 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 96. Giá trị của giới hạn 2 2 1 ... lim 1, 1 1 ... n n a a a ab b b b bằng: A. 0. B. 1 . 1 b a C. 1 . 1 a b D. Không tồn tại. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 97. Rút gọn 2 4 6 2 cos cos cos c s 1 o n S x x x x với cos 1. x A. 2 sin . Sx B. 2 cos . Sx C. 2 1 . sin S x D. 2 1 . cos S x Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 98. Rút gọn 2 4 6 2 1 sin si 1 n sin .sin n n S x x x x với sin 1. x A. 2 sin . Sx B. 2 cos . Sx C. 2 1 . 1 sin S x D. 2 tan . Sx Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 1. Giới Hạn Dãy số 53 Lớp Toán Thầy–Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 99. Thu gọn 23 1 tan tan tan S với 0. 4 A. 1 . 1 tan S B. cos . 2 sin 4 S C. tan . 1 tan S D. 2 tan . S Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 100. Cho , mn là các số thực thuộc 1;1 và các biểu thức: 23 1 M m m m 23 1 N n n n 2 2 3 3 1 A mn m n m n Khẳng định nào dưới đây đúng? A. . 1 MN A MN B. . 1 MN A MN C. 1 1 1 . A M N MN D. 1 1 1 . A M N MN Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 101. Số thập phân vô hạn tuần hoàn 0,5111 được biểu diễn bởi phân số tối giản a b . Tính tổng . T a b A. 17. B. 68. C. 133. D. 137. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV. Giới Hạn Dãy Số 54 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 102. Số thập phân vô hạn tuần hoàn 0,353535... A được biểu diễn bởi phân số tối giản a b . Tính . T ab A. 3456. B. 3465. C. 3645. D. 3546. 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Câu 103. Số thập phân vô hạn tuần hoàn 5,231231... B được biểu diễn bởi phân số tối giản a b . Tính . T a b A. 1409. B. 1490. C. 1049. D. 1940. 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Câu 104. Số thập phân vô hạn tuần hoàn 0,17232323 được biểu diễn bởi phân số tối giản a b . Khẳng định nào dưới đây đúng? A. 15 2. ab B. 14 2. ab C. 13 2. ab D. 12 2. ab Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 55 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 A. L THU T I. Giới hạn của hàm số tại một đi ểm 0 xx . 1. Giới hạn hữu hạn: Cho 0 ;, x a b f x là hàm số xác đ ịnh trên tập hợp: 0 ;\ D a b x , nếu với mọi dãy số 0 ;\ n x a b x sao cho 0 lim n xx , ta đ ều có: lim n f x L , thì lúc đó ta nói hàm số fx có giới hạn là L khi dần đ ến 0 x và đư ợc kí hiệu: 0 lim xx f x L . Chú ý: 0 0 lim xx xx 0 lim xx CC , (C : hằng số). Ví dụ 1. Tìm các giới hạn sau a). 2 1 lim 3 2 1 x xx . b). 3 2 2 31 lim 3 x x x x x . c). 2 2 2 3 10 lim . 3 5 2 x xx xx d). 32 3 2 3 9 2 lim 6 x x x x xx . e). 2 2 4 1 3 lim 4 x x x . f). 2 22 lim . 73 x x x h). 3 0 1 4 1 lim x x x . g). 3 2 3 2 1 3 2 4 2 lim 32 x x x x xx . k). 3 2 1 1 lim 32 x x x l). 0 9 16 7 lim x xx x . m). 3 32 0 1 3 1 1 lim x x x x xx . n). 322 0 3 1 2 1 lim 1 cos x xx x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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BÀI 2: GIỚI HẠN CỦA HÀM SỐ Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 56 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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.................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 2. Giới hạn vô cực: Cho 0 ;, x a b f x là hàm số xác đ ịnh trên tập hợp: Nếu với mọi dãy số n x với 0 ;\ n x a b x sao cho 0 lim n xx , ta đ ều có: lim n fx thì ta nói 0 lim xx fx . Nếu với mọi dãy số n x với 0 ;\ n x a b x sao cho 0 lim n xx , ta đ ều có: lim n fx thì ta nói 0 lim xx fx . II. Giới hạn của hàm số tại vô cực x 1. Các đ ịnh nghĩa: Cho fx là hàm số xác đ ịnh trên ; a , nếu với mọi dãy số n x với n x ; a và lim n x , ta đ ều có: lim n f x L thì ta nói lim x f x L Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 57 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Các giới hạn lim ; lim ; lim ; lim ; lim x x x x x f x f x f x L f x f x được định nghĩa hoàn toàn tương t ự. 2. Các giới hạn đ ặc biệt: lim k x x 2 lim 21 k x khi k n x khi k n lim x CC , (C là hằng số) lim 0 k x C x Ví dụ 2. Tìm các giới hạn sau a). 3 3 31 lim 31 x xx xx . b). 2 lim 1 x x x x . c). 2 lim 3 1 x x x x d). 3 23 31 lim 2 6 6 x xx xx . e). 32 1 lim 4 x x x xx . f). 2 31 lim 9 2 1 x x xx g). 5 3 1 lim 1 x xx x . h). 2 2 23 lim 4 1 2 x x x x xx . k). 20 30 50 2 3 3 2 lim 21 x xx x l). 2 lim 2 1 4 4 3 x x x x . m). 3 23 lim 1 1 x xx . n). 332 lim 3 1 2 x xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 58 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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.................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 3. Một số định lí về giới hạn Định lí 1: Nếu 0 lim xx x f x L và 0 lim xx x g x M thì: 0 lim xx x f x g x L M 0 lim xx x f x g x L M 0 lim . . xx x f x g x LM 0 lim . . xx x C f x CL , (C là hằng số); 0 0 lim . . kk xx C x C x ,(C là hằng số, k 0 lim 0 xx x fx L khi M g x M Định lí 2: Giả sử 0 lim xx x f x L 0 lim xx x f x L ; 0 3 3 lim xx x f x L ; Nếu 0 0 0 0, ; \ f x x x x x và 0 lim xx f x L thì 0 L và 0 lim xx f x L Ví dụ 3. Tìm các giới hạn sau a). 2 3 3 lim 6 x x xx . b). 4 3 2 2 2 3 2 lim 2 x xx xx . c). 2 2 2 32 lim 4 x xx x d). 2 2 2 5 lim 2 x xx x . e). 3 23 31 lim 2 x xx xx . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 59 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Định lí 3: Cho 3 hàm số ,, f x g x h x xác đ ịnh trên tập: D 0 0 0 ; \ , 0 x x x Nếu , f x g x h x x D và 00 lim lim x x x x g x h x L thì: 0 lim xx h x L . Từ đó ta ch ứng minh đư ợc: 0 0 0 sin sin lim 0 lim 1 lim 0 x x x x x x ux x khi u x x u x . Ví dụ 4. Tìm các giới hạn sau a). 0 sin 5 lim x x x . b). 0 tan 2 lim 3 x x x . c). 0 1 cos lim . sin x x x d). 2 0 1 cos lim x x x . e). 3 0 sin 5 .sin 3 .sin lim 45 x x x x x . f). 0 sin 7 sin 5 lim . sin x xx x g). 0 1 cos5 lim 1 cos3 x x x . h). 2 0 1 cos 2 lim .sin x x xx . k). 0 .sin lim 1 cos x x ax ax Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 60 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 61 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 3. Các quy tắc tìm giới hạn vô cực: Nếu 0 lim xx fx thì 0 1 lim 0 xx fx . Nếu 0 lim 0 xx f x L và 0 lim xx gx thì 0 0 0 ' ' lim r l m t ái d im . li âu xx xx xx và cùng d h â k u i L g x f x g x khi L g và x Nếu 0 lim 0 xx f x L và 0 lim 0 xx gx thì 0 0 0 ' ' lim t l l rái i d m im âu xx xx xx và cùng d h â k u i L g x fx gx khi L g và x Ví dụ 5 . Tìm các giới hạn sau a). 3 2 31 lim 31 x xx xx . b). 2 2 7 lim 1 2 3 1 x xx x x . c). 2 1 3 lim 42 x x x x xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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B. PHÂN DẠNG VÀ BÀI TẬP MINH HỌA. Dạng 1. Tìm giới hạn của hàm số bằng định nghĩa. 1. Phương pháp. a). Để tìm 0 lim xx fx ta làm như sau: Xét dãy số n x bất kỳ thuộc tập xác đ ịnh D với 0 n xx mà 0 lim n xx Tìm lim n fx : Nếu ta có lim n f x L thì 0 lim xx f x L . Nếu ta có lim n fx thì 0 lim xx fx . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 62 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 b). Để tìm lim x fx hoặc lim x fx ta làm như sau : Xét dãy số n x bất kỳ thuộc tập xác đ ịnh mà lim n x . Tìm lim n fx : Nếu ta có lim n f x L thì lim x f x L . Nếu ta có lim n fx thì lim x fx . Hoàn toàn tương t ự khi tính lim x fx . c). Để chứng minh hàm số fx không có giới hạn khi 0 xx ta thư ờng làm như sau : Chọn hai dãy số n u và n v cùng thuộc tập xác đ ịnh của hàm số sao cho 00 , nn u x v x và có 0 lim lim nn u v x . Chứng minh lim lim nn f u f v hoặc một trong hai giới hạn này không tồn tại. Khi đó theo đ ịnh nghĩa ta suy ra hàm s ố không có giới hạn khi 0 xx . Đối với các trư ờng hợp 00 , , , x x x x x x ta cũng làm tương t ự. 2. Bài tập minh họa Bài tập 1. Dùng định nghĩa tìm các giới hạn sau: a). 2 1 34 lim 1 x xx x . b). 2 2 1 4 lim 1 x x x . c). 2 lim 9 2 x xx . d). 2 2 lim 2 sin 4 x x x x . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 63 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 2. Dùng định nghĩa tìm giới hạn của hàm số a). 0 1 lim .sin x x x . b). 2 2 31 lim 1 x xx x . c). 2 2 1 23 lim 21 x xx xx . 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Bài tập 3 . Chứng minh rằng 1 3 lim sin 1 x x không tồn tại. 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 64 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Dạng 2. Tìm giới hạn của hàm số tại một điểm 0 xx bằng quy tắc, định lý : 0 lim xx fx 1. Phương pháp chung. Thay 0 xx vào fx nếu kết quả bằng một số C thì ta kết luận giới hạn đó b ằng C , tức là: 00 00 lim lim x x x x f x f x f x C Thay 0 xx vào fx nếu kết quả bằng 0 0 thì ta kết luận giới hạn vô đ ịnh, để khử nó ta xét 3 bài toán sau. Bài toán 1. Hàm số Px fx Qx trong đó , P x Q x là đa thức theo biến x . Nếu 0 lim 0 xx Px ; 0 lim 0 xx Qx thì 0 lim xx Px Qx được gọi là có dạng vô đ ịnh 0 0 . Ta biến đ ổi về dạng 01 01 . . m n x x P x x x Q x rồi giản ư ớc các thừa số có dạng 0 k xx ; max , k m n Px Qx Các cách biến đ ổi : Phân tích đa th ức thành nhân tử, sau đó rút g ọn biểu thức làm cả tử và mẫu bằng 0. Phân tích đa th ức thành nhân tử có các phương pháp sau Sử dụng bảy hằng đ ẳng thức đáng nh ớ. Nếu tam thức bậc hai thì sử dụng 2 12 ,0 ax bx c a x x x x a với 12 , xx là nghiệm của phương trình 2 0 ax bx c . Sử dụng phương pháp Hoocner . Phép chia đa th ức 4 3 2 P x ax bx cx dx e cho 0 () xx theo sơ đ ồ Hoocner: 2. Bài tập minh họa. Bài tập 4. Tìm các giới hạn sau: a). 3 2 2 8 lim 11 18 x x xx b). 32 32 3 2 5 2 3 lim 4 13 4 3 x x x x L x x x c). 32 32 1 2 5 4 1 lim 1 x x x x x x x d). 3 2 1 12 lim 28 x xx e). 3 42 1 1 lim 43 x x xx f). 22 2 11 lim 3 2 5 6 x x x x x Lời giải Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 65 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 66 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Bài toán 2. Hàm số Px fx Qx trong đó , P x Q x là các biểu thức có chứa căn thức theo x . 1. Phương pháp. Bước 1. Nếu biểu thức dư ới dấu giới hạn có chứa dấu căn: ta nhân và chia bi ểu thức liên hợp của biểu thức chứa căn ti ến về 0, rồi làm tương t ự như dạng trên ta sẽ khử được dạng vô đ ịnh. 22 22 22 ab ab ab a b a b a b ab ab ab 33 22 ab ab a ab b 33 22 ab ab a ab b . 2 2 3 3 3 3 3 22 22 3 3 3 3 . .. a b a ab b ab ab a ab b a ab b . 2 2 3 3 3 3 3 22 22 3 3 3 3 . .. a b a ab b ab ab a ab b a ab b 2 2 3 3 3 3 3 22 22 3 3 3 3 . .. a b a a b b ab ab a a b b a a b b 2 2 3 3 3 3 3 22 22 3 3 3 3 . .. a b a a b b ab ab a a b b a a b b 22 3 3 3 3 3 3 33 2 2 2 2 3 3 3 3 3 3 3 3 . .. a b a a b b ab ab a a b b a a b b . 22 3 3 3 3 3 3 33 2 2 2 2 3 3 3 3 3 3 3 3 . .. a b a a b b ab ab a a b b a a b b Bước 2: Phân tích đa th ức thành nhân tử, sau đó rút g ọn hạng tử chung của cả tử và mẫu. 2. Bài tập minh họa. Bài tập 5 . Tìm các giới hạn sau (khử căn bậc hai) : a). 1 32 lim 1 x x x b). 2 7 23 lim 49 x x x c). 22 2 3 2 6 2 6 lim 43 x x x x x xx d). 2 22 lim 73 x x x e). 2 4 1 21 lim x x x x xx f). 2 22 lim 13 x xx xx . g). 1 4 5 3 6 lim 32 x xx x h). 3 1 3 5 lim 2 3 6 x xx xx k). 2 2 0 11 lim 93 x x x l). 2 1 3 2 4 2 lim 1 x x x x x m). 2 2 0 42 lim 93 x x x n). 2 0 1 2 1 lim x x x x x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 67 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 68 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 6 . Tìm các giới hạn sau (khử căn bậc 3) : a). 3 2 42 lim 2 x x x b). 3 3 2 1 10 2 1 lim 32 x xx xx c). 3 3 2 3 27 lim 1 4 28 x x xx d). 3 3 1 1 lim 21 x x x e). 33 1 21 lim 1 x xx x f). 4 1 4 3 1 lim 1 x x x . g). 3 1 32 lim 1 x xx x h). 3 2 3 58 lim 2 x xx x k). 3 3 1 1 lim 4 4 2 x x x . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 69 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 70 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài toán 3: Thêm bớt số hạng hoặc một biểu thức vắng để khử được dạng vô định: (khử căn bậc hai và bậc ba) 1. Một số dạng thường gặp. 0 0 lim kk xx f x g x c xx hoặc 0 0 lim km xx f x g x c xx hoặc 0 0 lim km n xx f x g x c xx . Trong đó * ,, k m n N và min( , ) n k m . 2. Phương pháp : Bài toán 1. 0 ( ) ( ) lim nm xx f x g x x Ta tách về tổng hoặc hiệu của hai lim : việc tách hằng số C (biến số) đư ợc làm như sau. Gi ả sử ta tính giới hạn: 0 ( ) ( ) lim nm xx f x g x x . Ta tiến hành tách ra: 0 0 0 12 ( ) ( ) ( ) ( ) lim lim lim n m n m x x x x x x f x c c g x f x c c g x II x x x . Tìm C bằng cách: thế 0 xx vào: 00 00 ( ) 0 ( ) ( ) 0 ( ) nn mm f x c c f x c f x c c f x Nếu giới hạn có dạng tổng quát : 0 1 ( ) ( ) lim .... nm nn xx f x g x ax bx c . Ta tiến hành tách ra: 0 1 2 1 2 1 ( ) ( ... ) ( ... ) ( ) lim .... n n n n nm nn xx f x x x x x g x ax bx c Sử dụng phương pháp đ ồng nhất thức. 3. Bài tập minh họa. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 71 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Bài tập 7 . Tìm các giới hạn sau : a). 1 2 2 5 4 5 lim 1 x xx x b). 3 2 3 2 5 6 lim 2 x xx x c). 3 2 2 2 2 4 11 7 lim 4 x x x x x d). 3 2 3 2 3 2 lim 2 x xx x e). 22 1 5 1 3 1 5 2 1 lim 1 x x x x x x f). 3 32 2 1 57 lim 1 x xx x g). 3 0 2 1 8 lim x xx x h). 3 0 3 8 5 4 lim x xx x k). 2 3 2 1 3 2 4 2 lim 32 x x x x xx l). 3 2 2 8 11 7 lim 2 5 2 x xx xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 72 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 73 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 74 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 8. Tính các giới hạn sau: a). 3 2 0 1 4 1 6 lim x xx x b). 3 22 2 2 2 6 5 3 9 7 lim 2 x x x x x x c). 3 3 2 2 3 0 8 6 9 9 27 27 lim x x x x x x x d). 3 2 32 1 6 2 2 lim 1 x xx x x x e). 2 2 2 2 1 3 2 4 19 3 46 lim 1 x x x x x x f). 3 3 2 2 3 4 24 2 8 2 3 lim 4 x x x x x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 75 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 76 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 77 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 9. Tìm các giới hạn sau a) . 3 4 1 1 lim 1 x x x b). 3 2 4 2 0 1 1 2 lim x xx xx . c) . 4 3 1 21 lim 21 x x x d) . 3 4 2 6 7 2 lim 2 x xx x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài toán 2. Tính 0 0 ( ) 1 ( ) lim n xx P x ax Q x xx bằng cách thêm bớt Px rồi đ ặt nhân tử chung. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 2. Giới H ạn Hàm Số 78 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Tách 00 00 ( ) 1 ( ) ( ) ( ) ( ) 1 ( ) lim lim n n x x x x P x ax P x Q x P x P x ax Q x x x x x 00 12 00 ( ) P( ) ( ) 1 ( ) lim lim n x x x x Q x x P x ax P x II x x x x Tính hai 1 I và 2 . I Bài tập 10. Tính các giới hạn sau: a). 0 1 4 . 1 6 1 lim x xx x b). 3 0 4 . 1 2 2 lim x xx x c). 3 1 3 1 2 2 lim 1 x xx x d). 3 2 0 4 8 3 4 lim x xx xx e). 3 4 0 1.2 1 2.3 1 3.4 1 1 lim x xxx x f). Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 79 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 4. Câu hỏi trắc nghiệm Mức độ 1. Nhận biết Câu 1. Giá trị của giới hạn 2 2 lim 3 7 11 x xx là: A. 37. B. 38. C. 39. D. 40. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 2. Giá trị của giới hạn 2 3 lim 4 x x là: A. 0. B. 1. C. 2. D. 3. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 3. Giá trị của giới hạn 2 0 1 lim sin 2 x x là: A. 1 sin . 2 B. . C. . D. 0. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 4. Giá trị của giới hạn 2 3 1 3 lim 2 x x x là: A. 1. B. 2. C. 2. D. 3 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 5. Giá trị của giới hạn 3 4 1 lim 2 1 3 x xx xx là: A. 1. B. 2. C. 0. D. 3 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 6. Giá trị của giới hạn 4 1 1 lim 3 x x xx là: A. 3 . 2 B. 2 . 3 C. 3 . 2 D. 2 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 7. Giá trị của giới hạn 2 1 31 lim 1 x xx x là: Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 80 A. 3 . 2 B. 1 . 2 C. 1 . 2 D. 3 . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 8. Giá trị của giới hạn 2 4 3 9 lim 2 1 3 x xx xx là: A. 1 . 5 B. 5. C. 1 . 5 D. 5. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 9. Giá trị của giới hạn 2 3 2 2 1 lim 2 x xx xx là: A. 1 . 4 B. 1 . 2 C. 1 . 3 D. 1 . 5 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 10. Giá trị của giới hạn 3 2 2 3 4 3 2 lim 1 x xx x là: A. 3 . 2 B. 2 . 3 C. 0. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Mức độ 2. Thông Hiểu Câu 11. Giá trị của giới hạn 3 2 2 8 lim 4 x x x là: A. 0. B. . C. 3. D. Không xác định. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 12. Giá trị của giới hạn 5 3 1 1 lim 1 x x x là: A. 3 . 5 B. 3 . 5 C. 5 . 3 D. 5 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 13. Giá trị của giới hạn 2 2 3 6 lim 3 x xx xx là: Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 81 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 A. 1 . 3 B. 2 . 3 C. 5 . 3 D. 3 . 5 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 14.(Sở Quảng Ninh Lần1) 2018 2 2018 2018 2 4 lim 2 x x x A. 2019 2 B. 2018 2 C. 2 D. . 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Câu 15. Biết rằng 3 2 3 2 6 3 lim 3 . 3 x x ab x Tính 22 . ab A. 10. B. 25. C. 5. D. 13. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 16. (THPT Lý Thái Tổ) Tính giới hạn 2 2 2 2 3 3 lim 4 x xx L x . A. 2 7 L . B. 7 24 L . C. 9 31 L . D. 0 L . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 17. Giá trị của giới hạn 2 21 21 7 0 12 lim x xx x là: A. 21 2 . 7 B. 21 2 . 9 C. 21 2 . 5 D. 21 12 . 7 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 82 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 18.(THPT Hoàng Hoa Thám) Tính 2 2 28 lim 2 5 1 x xx x . A. 3 . B. 1 2 . C. 6 . D. 8 . 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Câu 19.(Tạp Chí Toán Học)Giới hạn 3 1 5 1 lim 43 x xx xx bằng a b (Phân số tối giản). Giá trị thực của ab là A. 1 . B. 1 9 . C. 1 . D. 9 8 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 20. (Tạp Chí Toán Học)Giới hạn 1 2 7 2 lim 54 x xx xx bằng a b (Phân số tối giản). Giá trị thực của ab là A. 10 . B. 1 9 . C. 8 . D. 10 9 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 83 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Mức độ 3. Vận dụng Câu 21. Giá trị của giới hạn 3 3 1 1 lim 4 4 2 x x x là: A. 1. B. 0. C. 1. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 22. Giá trị của giới hạn 3 0 2 1 8 lim x xx x là: A. 5 . 6 B. 13 . 12 C. 11 . 12 D. 13 . 12 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 23. Biết rằng 0, 5 b a b và 3 0 11 lim 2 x ax bx x . Khẳng định nào dưới đây sai? A. 1 3. a B. 1. b C. 22 10. ab D. 0. ab Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 84 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 24. (THPT Lý Thái Tổ) Tìm tất cả các giá trị của tham số m để 2 B với 32 1 lim 2 2 5 5 x B x x m m . A. 0;3 m . B. 1 2 m hoặc 2 m . C. 1 2 2 m . D. 23 m . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 25.(THPT Lý Thái Tổ) Nếu 2 lim 5 x fx thì 2 lim 3 4 x fx bằng bao nhiêu? A. 18 . B. 1 . C. 1. D. 17 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 26.(THPT Gia Lộc Hải Dương 2019) Cho hàm số y f x thỏa mãn: 2 1 3 5 2 2 1 xx f xx 1 2; 2 xx . Tìm lim ? x fx A. 4 3 . B. 1 5 . C. 3 2 . D. 2 3 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 27.(THPT Sơn Tây Hà Nội 2019) Cho hàm số y f x có đạo hàm tại điểm 0 2 x . Tính 2 22 lim 2 x f x xf x . A. 2 2 2 ff . B. 0. C. 2 f . D. 2 2 2 ff . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 85 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 28.(THPT Lý Thái Tổ) Cho 1 1 lim 1 1 x fx x . Tính 2 1 2 I lim 1 x x x f x x A. 5 I . B. 4 I . C. 4 I . D. 5 I . 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Câu 29. (HSG Lớp 12 Bắc Giang) Choa ,b là các số thực dương thỏa mãn 8 ab và 2 0 2 1 1 lim 5 x x ax bx x . Trong các mệnh đề dưới đây, mệnh đề nào đúng? A. 2;4 a . B. 3;8 a . C. 3;5 b . D. 4;9 b . 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Câu 30. (Kênh truyền Hình GD Quốc Gia 2019) Cho , mn là các số thực khác 0 . Nếu giới hạn 2 1 lim 3 1 x x mx n x thì . mn bằng A. 3 . B. 1 . C. 3 . D. 2 . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 86 Câu 31. (Sở GD Và Đào Tạo Vĩnh Phúc) Tính 2 3 1 21 lim 1 x x a x a x . 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Câu 32. Biết rằng 4 ab và 3 1 lim 11 x ab xx hữu hạn. Tính giới hạn 3 1 lim 11 x ba L xx . A. 1. B. 2. C. 1. D. 2. 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Câu 33. Kết quả của giới hạn 0 1 lim 1 x x x là: A. . B. 1. C. 0. D. . 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Câu 34. Kết quả của giới hạn 2 2 0 1 lim sin x xx x là: A. 0 . B. 1 . C. . D. . 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Câu 35. (THPT Chuyên Bắc Giang) Cho biết 2 3 1 2 12 lim 4 3 1 x ax bx c xx với ,, abc . Tập nghiệm của phương trình 42 2 2 0 ax bx c trên có số phần tử là A. 1. B. 3 . C. 0 . D. 2 . Lời giải Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 87 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Câu 36.(Chuyên KHTN 2019) Cho hàm số y f x xác định trên thỏa mãn 2 16 lim 12 2 x fx x . Tính giới hạn 3 2 2 5 16 4 lim 28 x fx xx . A. 5 24 . B. 1 5 . C. 5 12 . D. 1 4 . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 88 Câu 37. Tính 2018 2018 2 2 2018 1 1 2 2.3 lim 1 2017 x x x x xx A. 2017 5.3 . B. 2017 3 . C. 2017 8.3 . D. 2017 2.3 . 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Dạng 3. Tìm giới hạn của hàm số khi x 1. Phương pháp. Để tìm lim x fx ta làm như sau: Chia tử và mẫu cho k x với k x là lũy thừa có số mũ lớn nhất của tử và mẫu, (hoặc rút k x làm nhân tử) sau đó áp dụng các định lý về giới hạn hữu hạn hoặc các quy tắc về giới hạn vô cực. Nếu lim x f x C thì kết luận hàm số đó có giới hạn là . C Nếu lim lim . xx f x P x Q x với lim , lim xx P x L Q x thì ta dựa vào quy tắc dấu để kết luận kết quả của giới hạn là hoặc . Nếu lim lim xx Px fx Qx với đa thức , P x Q x có bậc lần lượt mà , mn và lim , lim 0 xx P x L Q x ta đặt mũ cao nhất của tử và mẫu rồi tách ra cùng bậc đưa về lim của tích. Nếu gặp dạng vô định ,0 ; thì ta khử dạng vô định: Dạng : Nếu 0 lim xx x fx ; 0 lim xx x gx thì 0 lim xx x fx gx được gọi là có dạng vô định. Dạng 0 : Nếu 0 lim 0 xx x fx ; 0 lim xx x gx thì 0 lim . xx x f x g x được gọi là có dạng vô định . Dạng : Nếu 0 lim xx x fx ; 0 lim xx x gx thì 0 lim xx x f x g x được gọi là có dạng vô định . Khử dạng vô định bằng phương pháp nhân lượng liên hợp. 2. Bài tập minh họa Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 89 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Bài toán 1. Giới hạn hữu hạn lim lim . xx f x P x Q x với lim , lim xx P x L Q x Phương pháp: Đặt mũ cao nhất rồi dựa vào quy tắc dấu để kết luận kết quả của giới hạn là hoặc . Bài tập 11. Tính giới các giới hạn sau: a). 3 lim (2 3 ) x xx b). 2 lim 3 4 x xx c). 2 lim 2 1 x xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài toán 2. Giới hạn hữu hạn hữu tỉ lim lim xx Px fx Qx ( bậc tử bé hơn hoặc bằng bậc mẫu) Phương pháp. ta đặt mũ cao nhất của tử và mẫu rồi rút gọn nhân tử chung. Bài tập 12. Tìm các giới hạn sau: a). 2 3 2 3 6 lim 4 3 5 x xx xx . b). 4 6 7 5 9 3 2 3 5 3 6 2 lim 3 2 6 3 7 2 x x x x x x x . c). 2 2 4 2 1 2 lim 9 3 2 x x x x x x x d). 2 2 2 3 4 1 lim 4 1 2 x x x x xx . e). 2 2 23 lim 4 1 2 x x x x xx . f). 2 1 lim 1 x xx xx g). 2 2 2 1 3 lim 5 x xx xx . h). 3 3 2 5 lim x x x x x x x . k). 53 3 23 21 lim 21 x xx x x x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 90 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 13. Tìm giới hạn của các hàm số sau: a). 2 2 3 2 1 lim 5 1 2 x xx x x x b). 42 lim 1 21 x x x xx c). 2 1 lim 1 x xx xx d). 2 23 lim 5 x x L xx e). 3 52 2 lim . 3 x xx x xx f). 42 21 lim 12 x xx x . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 91 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 14. Tính các giới hạn sau: a). 2 lim 10 x x x x x b). 2 32 lim 31 x x x x x c). 2 2 2 3 1 lim 4 1 1 x x x x xx d). 2 2 2 3 1 lim 4 1 1 x x x x xx e). 3 3 2 2 3 2 2 3 2 ( 2 ) 2 lim 32 x x x x x x x xx f). 2 3 23 lim x xx x xx g). 42 4 4 1 2 4 lim 2 x xx x x x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 92 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài toán 3. Giới hạn vô cực (bậc tử lớn hơn bậc mẫu: ) lim lim 0 xx L Px fx Qx Phương pháp. Ta đặt mũ cao nhất của tử và mẫu rồi tách ra cùng bậc đưa về lim của tích 0 lim . xx x f x g x Bài tập 15. Tìm các giới hạn sau: a). 2 52 lim 21 x xx x . b). 42 21 lim 12 x xx x . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 93 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài toán 4. Giới hạn vô cực dạng vô định 0 1. Phương pháp. Khi ta đặt mũ cao nhất của tử và mẫu thì xuất hiện ; 0; Khử dạng vô định bằng phương pháp nhân lượng liên hợp rồi làm bình thường bài toán 1,2. 2. Bài tập minh họa Bài tập 16. Tìm các giới hạn sau: a). 2 lim ( ) x x x x . b). 2 lim ( 3 2 ) x x x x . c). 2 lim ( ) x x x x d). 2 lim ( 3 2 ) x x x x . e). 2 lim ( 3 2 ) x x x x . f). lim ( 2 2) x xx g). 22 lim 4 3 3 2 x x x x x . h). 22 4 3 3 2 x lim x x x x . k). 4 2 2 lim 4 3 1 2 x x x x l). 3 3 lim 8 1 2 1 x xx m). 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 94 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 17. Tìm các giới hạn sau a). 22 lim 1 2 x x x x . b). lim 3 3 3 3 x x x x x . c). 332 lim 6 x x x x d). 332 lim 3 1 2 x xx e). lim 2 2 1 x x x x f). 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 95 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài toán 5. Giới hạn vô cực dạng vô định 0 1. Phương pháp. Giả sử cần tìm giới hạn của hàm số . h x f x g x khi 0 xx hoặc x trong đó 0 fx và gx . Ta thường biến đổi theo các hướng sau: Nếu 0 xx thì ta thường viết . 1 fx f x g x gx sẽ đưa về dạng vô định 0 0 . Nếu x thì ta thường viết . 1 gx f x g x fx sẽ đưa về về dạng . Tuy nhiên ở nhiều bài toán giới hạn loại này ta chỉ cần thực hiện một số biến đổi như đưa thừa số vào trong dấu căn thức, quy đồng mẫu số,... ta có thể đưa về giới hạn quen thuộc. 2. Bài tập minh họa Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 96 Bài tập 18. Tìm các giới hạn sau: a). 3 3 1 1 1 lim 3 3 x x x . b). 3 1 lim 2 x x x xx . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Dạng 4. Tìm giới hạn của hàm số các hàm đặc biệt. 1. Phương pháp. 1 1 1 1 n n 3 3 2 3 3 so hang a b a b a ab b 1 2 3 2 2 1 so hang n n n n n n n n a b a b a a b a b ab b 2. Bài tập minh họa Bài tập 19. Dùng định nghĩa tìm các giới hạn sau: a). 2 1 ... lim 1 n x x x x n x . b). 2 1 1 lim 1 n x x nx n x . c). 0 11 lim n x ax L x . d). 0 11 lim nm x ax bx L x . e). 0 11 lim 0 11 n m x ax L ab bx f). 0 11 lim 11 nm x ax bx L x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 97 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 20. Dùng định nghĩa tìm các giới hạn sau: a). 1 1 lim 1 m n x x L x . b). 23 23 1 lim n m x x x x x n L x x x x m . c). 100 50 1 21 lim 21 x xx L xx . d). 1 2 1 1 lim 1 n x x n x n L x . e). 1 lim , , * 11 mn x mn mn xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 98 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Câu hỏi trắc nghiệm Mức độ 1. Nhận biết Câu 38. Giá trị của giới hạn 3 lim 1 x xx là: A. 1. B. . C. 0. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 39. Giá trị của giới hạn 3 2 lim 2 3 x x x x là: A. 0. B. . C. 1. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 40. Giá trị của giới hạn 2 lim 1 x xx là: A. 0. B. . C. 2 1. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 41. Giá trị của giới hạn 332 lim 3 1 2 x xx là: A. 3 3 1. B. . C. 3 3 1. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 99 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Câu 42. Giá trị của giới hạn 2 lim 4 7 2 x x x x x là: A. 4. B. . C. 6. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 43. Kết quả của giới hạn 2 2 2 5 3 lim 63 x xx xx là: A. 2. B. . C. 3. D. 2 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 44. Kết quả của giới hạn 32 2 2 5 3 lim 63 x xx xx là: A. 2. B. . C. . D. 2 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 45. Kết quả của giới hạn 32 65 2 7 11 lim 3 2 5 x xx xx là: A. 2. B. . C. 0. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 46. Kết quả của giới hạn 2 23 lim 1 x x xx là: A. 2. B. . C. 3. D. 1 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 47. (THPT Lý Thái Tổ) Trong các mệnh đề sau mệnh đề nào sai? A. 2 1 lim 1 2 x x x x . B. 2 1 2 1 lim 2 3 2 x xx x . C. 1 32 lim 1 x x x . D. 32 lim 3 2 x x x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 0 Mức độ 2. Thông Hiểu Câu 48. Kết quả của giới hạn 2 41 lim 1 x xx x là: A. 2. B. 1. C. 2. D. . 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Câu 49. Kết quả của giới hạn 2 2 4 2 1 2 lim 9 3 2 x x x x x x x là: A. 1 . 5 B. . C. . D. 1 5 . 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Câu 50. Kết quả của giới hạn 332 2 21 lim 21 x xx x là: A. 2 . 2 B. 0. C. 2 . 2 D. 1. 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Câu 51. Giá trị của giới hạn 32 lim 2 x xx là: A. 1. B. . C. 1. D. . 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Câu 52. Giá trị của giới hạn 2 lim 1 2 x xx là: A. 0. B. . C. 2 1. D. . 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Câu 53. Giá trị của giới hạn 2 lim 1 x xx là: A. 0. B. . C. 1 . 2 D. . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 1 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 54.(THPT Đoàn Thượng) Tính 2 23 lim 1 x x xx . A. 0 . B. . C. 1 . D. 1. 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Câu 55.( THPT Lý Thái Tổ) Giá trị 2 3 6 2 23 lim x x x x x bằng A. 1 2 . B. 9 17 . C. 3 2 . D. 1. 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Mức độ 3. Vận dụng Câu 56. Giá trị của giới hạn 22 lim 3 4 x x x x x là: A. 7 . 2 B. 1 . 2 C. . D. . 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Câu 57. Giá trị của giới hạn 332 lim 3 1 2 x xx là: A. 3 3 1. B. . C. 3 3 1. D. . 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Câu 58. Giá trị của giới hạn 33 lim 2 1 2 1 x xx là: A. 0. B. . C. 1. D. . Lời giải. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 2 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 59. Kết quả của giới hạn 32 21 lim 32 x x x xx là: A. 2 . 3 B. 6 . 3 C. . D. . 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Câu 60. Giá trị của giới hạn 3 2 3 2 lim x x x x x là: A. 5 . 6 B. . C. 1. D. . 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Câu 61. (THPT Chuyên Vĩnh Phúc 2019) 3 2 3 2 lim 2 3 x x x x x x A. 1 2 . B. 0 . C. . D. . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 3 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 62. Tìm tất cả các giá trị của a để 2 lim 2 1 x x ax là . A. 2. a B. 2. a C. 2. a D. 2. a Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 63. Biết rằng 2 2 4 2 1 2 lim 0 3 x x x x L ax x bx là hữu hạn (với , ab là tham số). Khẳng định nào dưới đây đúng. A. 0. a B. 3 . L ab C. 3 . L ba D. 0. b Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 64. Biết rằng 2 23 1 ax xx có giới hạn là khi x (với a là tham số). Tính giá trị nhỏ nhất của 2 2 4. P a a A. min 1. P B. min 3. P C. min 4. P D. min 5. P Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 65. Biết rằng 2 lim 5 2 5 5 . x x x x a b Tính 5. S a b A. 1. S B. 1. S C. 5. S D. 5. S Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 4 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 66.(THPT Lý Thái Tổ) Cho biết 2 1 4 5 2 lim 23 x xx ax . Giá trị của a bằng A. 3. B. 2 3 . C. 3 . D. 4 3 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 67.(Sở GD và Đào Tạo Cần Thơ 2018) Cho biết 2 4 7 12 2 lim 17 3 x xx ax . Giá trị của a bằng A. 3 . B. 3 . C. 6 . D. 6 . 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Câu 68.(THPT Lý Thái Tổ) Cho 3 23 2 2 2 5 1 lim 1 x x x x x a xb ( a b là phân số tối giản, , ab là số nguyên). Tính tổng 22 L a b . A. 150. B. 143. C. 140. D. 145. Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 5 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 69.(THPT Lý Thái Tổ Bắc Ninh) Có bao nhiêu giá trị m nguyên thuộc đoạn 20;20 để 2 lim 2 3 x mx m x A.21. B.22. C.20. D.41. Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 70. (THPT Lương Thế Vinh Hà Nội) Biết rằng 2 1 lim 5 2 x x ax b x . Tính tổng ab . A. 6 . B. 7 . C. 8 . D. 5 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 71. (THPT Lương Thế Vinh Hà Nội) Biết rằng 2 lim 2 3 1 2 2 x a x x x b , (a là số nguyên, b là số nguyên dương, a b tối giản). Tổng ab có giá trị là A. 1. B. 5 . C. 4 . D. 7 . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 6 Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 72. (HSG Bắc Ninh) Cho 2 lim 5 5 x x ax x . Khi đó giá trị a là A. 10 . B. 6 . C. 6 . D. 10 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 73.(THPT Nghèn Hà Tĩnh 2018) Biết 2 lim 4 3 1 0 x x x ax b . Tính 4 ab ta được A. 3 . B.5 . C. 1 . D. 2 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 74.(Sở GD và Đào Tạo Cần Thơ 2018) Cho 2 lim 5 5 x x ax x . Khi đó giá trị a là A. 6 . B. 10. C. 10 . D. 6 . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 7 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 75.(THPT Chuyên Thái Bình) Cho hai số thực a và b thoả mãn 2 4 3 1 lim 0 21 x xx ax b x . Khi đó 2 ab bằng: A. 4 . B. 5 . C. 4 . D. 3 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 76. (THPT Chuyên Vĩnh Phúc 2018) Cho 2 lim 5 5 x x ax x thì giá trị của a là một nghiệm của phương trình nào trong các phương trình sau? A. 2 11 10 0 xx . B. 2 5 6 0 xx . C. 2 8 15 0 xx . D. 2 9 10 0 xx . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 77.(Chuyên Lương Thế Vinh 2019) Giá trị của số thực m sao cho 2 3 2 1 3 lim 6 47 x x mx xx là A. 3 m . B. 3 m . C. 2 m . D. 2 m . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 8 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 78.(THPT Quãng Xương 2019) Cho 2 1 2017 1 lim 2018 2 x ax x ; 2 lim 1 2 x x bx x . Tính 4 P a b . A. 3 P . B. 1 P . C. 2 P . D. 1 P . 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Câu 79.(THPT Xuân Trường 2018) Cho số thực a thỏa mãn 2 2 3 2017 1 lim 2 2018 2 x ax x . Khi đó giá trị của a là A. 2 2 a . B. 2 2 a . C. 1 2 a . D. 1 2 a . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 80.(Tạp chí Toán Học) Tìm giá trị dương của k để 2 3 1 1 lim 9 2 x kx f x với 2 ln 5 f x x : A. 12 k . B. 2 k . C. 5 k . D. 9 k . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 0 9 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 81.(THPT Lý Thái Tổ) Cho , ab là các số dương. Biết 3 2 3 2 7 lim 9 27 5 27 x x ax x bx . Tìm giá trị lớn nhất của . ab A. 49 18 . B. 59 34 . C. 43 58 . D. 75 68 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Câu 82.(THPT Triệu Thị Trinh) Biết 2 3 1 2 7 1 2 lim 21 x x x x a c b x với a , b , c và a b là phân số tối giản. Giá trị của abc bằng: A. 5 . B. 37 . C. 13 . D. 51. Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 110 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 83.(THPT Yên Định 2018) Cho fx là một đa thức thỏa mãn 1 16 lim 24 1 x fx x . Tính 1 16 lim 1 2 4 6 x fx I x f x A. 24. B. I . C. 2 I . D. 0 I . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 84.(THPT Chuyên Lương Thế Vinh 2018) Cho 1 10 lim 5 1 x fx x . Giới hạn 1 10 lim 1 4 9 3 x fx x f x bằng A. 1. B. 2 . C. 10 . D. 5 3 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 2. Giới Hạn Hàm Số 1 1 1 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 112 A. LÝ THUYẾT I. CÁC ĐỊNH NGHĨA: 1. Giới hạn bên phải: Giả sử fx là hàm số xác định trên khoảng 0 ; xb , nếu với mọi dãy số n x với 0 n xx và lim 0 n xx , ta đều có: lim n f x L , thì lúc đó ta nói hàm số fx có giới hạn bên phải là số thực L khi dần đến 0 x và được Kí hiệu: 0 lim xx f x L Ví dụ 1. Tìm các giới hạn sau a). 2 2 31 lim 2 x xx x . b). 2 1 1 3 2 lim 1 x xx x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 2. Giới hạn bên trái: Giả sử fx là hàm số xác định trên khoảng 0 ; ax , nếu với mọi dãy số n x với 0 n xx và lim 0 n xx , ta đều có: lim n f x L , thì lúc đó ta nói hàm số fx có giới hạn bên trái là số thực L khi dần đến 0 x và được Kí hiệu: 0 lim xx f x L Ví dụ 2. Tìm các giới hạn sau a) 2 15 lim 2 x x x . b). 2 3 1 3 2 lim 3 x xx x . c). 2 5 5 lim 25 x x x . d). 2 5 5 lim 25 x x x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... BÀI 3: GIỚI HẠN MỘT BÊN Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 113 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 3. Giới hạn vô cực: Giả sử fx là hàm số xác định trên khoảng 0 ; xb , nếu với mọi dãy số x n với 0 n xx và 0 lim n xx , ta đều có: lim n x f , thì lúc đó ta nói hàm số fx có giới hạn bên phải là vô cực khi dần đến 0 x và được Kí hiệu: 0 lim xx fx Các định nghĩa 0 0 0 lim , lim , lim , x x x x x x f x f x f x được phát biểu tương tự trên. II. ĐỊNH LÝ VỀ SỰ TỒN TẠI CỦA GIỚI HẠN. Nếu 0 lim xx f x L thì hàm số f có giới hạn bên phải và giới hạn bên trái tại điểm 0 x . và 00 lim lim x x x x f x f x L Ngược lại, nếu 00 lim lim x x x x f x f x L thì hàm số f có giới hạn tại điểm 0 x và 0 lim xx f x L . Tương tự, 0 00 lim lim lim xx x x x x f x f x f x 0 00 lim lim lim xx x x x x f x f x f x Ví dụ 3. Cho hàm số 3 3 2 2 1 3 1 x x x fx x x x Tìm 11 lim ; lim . xx f x f x Hàm số có giới hạn tại 1 x không? Vì sao? Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... B. PHÂN DẠNG VÀ BÀI TẬP MINH HỌA. Dạng 1. Tìm giới hạn của hàm số bằng định nghĩa. 1. Phương pháp. Khi 0 xx thì 0 xx suy ra 0 2 00 0 xx x x x x x x x Khi 0 xx thì 0 xx suy ra 0 2 00 0 xx x x x x x x x 2. Bài tập minh họa. Bài tập 1. Tìm các giới hạn sau: Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 114 a). 3 3 lim 5 15 x x x b). 0 lim . x xx xx c). 0 2 lim x xx xx d). 2 32 1 43 lim x xx xx e). 3 2 1 44 lim 35 x x xx f). 0 2 lim x xx xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 2. Tìm các giới hạn sau: a). 2 2 4 lim 2 x x x b). 2 54 1 32 lim x xx xx c). 2 2 3 7 12 lim 9 x xx x d). 22 2 11 lim 3 2 5 6 x x x x x e). 2 2 2 lim 2 5 2 x x xx f). 2 2 32 lim 2 x xx x g). 2 3 1 3 2 lim 3 x xx x h). 2 2 3 2 5 3 lim 3 x xx x k). 2 2 11 lim 24 x xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 115 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 3. Tìm các giới hạn sau( đưa ra đa thứckhỏi căn hoặc đưa vào căn) a). 3 2 1 lim 1 1 x x x x b). 1 1 lim . 2 1 1 x x x xx c). 2 2 lim 2 4 x x x x d). 2 2 4 lim 2 x x x e). 2 1 2 2 2 5 2 lim 1 4 x xx x f). 2 1 5 lim 1 23 x x x xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 1 1 6 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 3. Câu hỏi trắc nghiệm. Câu 1. Kết quả của giới hạn 2 15 lim 2 x x x là: A. . B. . C. 15 . 2 D. 1. 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Câu 2.(THPT Lê Quý Đôn) Tính giới hạn 2 1 1 lim 1 x x x . A. 0 . B. . C. . D. 1. 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Câu 3.(THPT Chuyên Biên Hòa 2018) Tính giới hạn 2 2 3 lim 2 x x x . A. . B. 2 . C. . D. 3 2 . 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Câu 4.(THPT Nguyễn Trãi-Đà Nẵng-lần 1 năm 2017-2018) Tìm giới hạn 1 43 lim 1 x x x A. . B. 2 . C. . D. 2 . 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Câu 5.(THPT Chuyên Lương Thế Vinh) Kết quả của giới hạn 2 4 2 28 lim 2 x xx x là A. . B. 0 . C. . D.1. Lời giải Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 1 1 7 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 6.(THPT Chuyên Biên Hòa 2018) Tính giới hạn 2 2 3 lim 2 x x x . A. . B. 2 . C. . D. 3 2 . 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Câu 7. Kết quả của giới hạn 2 2 lim 2 x x x là: A. . B. . C. 15 . 2 D. Không xác định. 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Câu 8. Kết quả của giới hạn 2 36 lim 2 x x x là: A. . B. 3. C. . D. Không xác định. 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Câu 9. Kết quả của giới hạn 2 2 2 lim 2 5 2 x x xx là: A. . B. . C. 1 . 3 D. 1 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 10. Kết quả của giới hạn 2 2 3 13 30 lim 35 x xx xx là: A. 2. B. 2. C. 0. D. 2 . 15 Lời giải. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 118 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 11. Giá trị của giới hạn 3 3 3 lim 27 x x x là: A. 1 . 3 B. 0. C. 5 . 3 D. 3 . 5 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 12. Giá trị của giới hạn 2 2 0 lim x x x x x là: A. 0. B. . C. 1. D. . 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Câu 13. Giá trị của giới hạn 2 2 11 lim 24 x xx là: A. . B. . C. 0. D. 1. 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Câu 14. Kết quả của giới hạn 2 2 lim 2 4 x x x x là: A. 1. B. . C. 0. D. . 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Câu 15. Kết quả của giới hạn 3 2 1 lim 1 1 x x x x là: A. 3. B. . C. 0. D. . Lời giải. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 1 1 9 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 16.(Chuyên KHTN) Trong các mệnh đề sau, mệnh đề nào sai? A. 2 3 lim 1 2 2 x x x x . B. ( 1) 32 lim 1 x x x . C. 2 lim 1 2 x x x x . D. ( 1) 32 lim 1 x x x . 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Câu 17.(THPT Hoàng Hoa Thám 2018) Trong các mệnh đề sau, mệnh đề nào sai? A. 0 1 lim x x . B. 0 1 lim x x . C. 5 0 1 lim x x . D. 0 1 lim x x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 18. (THPT Trần Nhân Tông 2018) Trong các mệnh đề sau mệnh đề nào sai A. 2 3 lim 1 2 2 x x x x . B. 2 lim 1 2 x x x x . C. 1 32 lim 1 x x x . D. 1 32 lim 1 x x x . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 1 2 0 Dạng 2. Chứng minh sự tồn tại của giới hạn. 1. Phương pháp. Hàm số có giới hạn là L thì 0 00 lim lim lim xx x x x x f x L f x f x L Hàm số có giới hạn là thì 0 00 lim lim lim xx x x x x f x f x f x Hàm số có giới hạn là thì 0 00 lim lim lim xx x x x x f x f x f x 2. Bài tập minh họa. Bài tập 4. Cho hàm số 42 3 5 6 1 3 1 x x x x fx x x x Tìm 11 lim ; lim . xx f x f x Hàm số có giới hạn tại 1 x không? Vì sao? 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Bài tập 5. Cho hàm số 2 23 1 1 1 1 8 x x x y f x x a). Tìm 1 lim . x fx So sánh 1 lim x fx và 1 f b). Tìm 3 lim . x fx So sánh 3 lim x fx và 3. f Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 6. Cho hàm số 2 2 3 2 5 2 3 1 2 xx f x x xx a). Tìm 22 lim ; lim . xx f x f x b). Hàm số có giới hạn tại 2 x không? Tại sao? 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 1 2 1 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Cho hàm số 3 2 2 1 8 0 1 2 0 4 2 2 xx x x f x ax b x x x x Tìm a, b để hàm số cùng có giới hạn tại 2 x và 0. x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 8. Tìm các giới hạn sau : a). 1 lim x fx với 2 3, 1 13, 1 1 7 2, 1 x khi x f x x khi x x khi x b). 2 lim x gx với 32 , 2 1 10, 2 x khi x gx x x khi x Lời giải Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 1 2 2 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 9. Tìm giới hạn của hàm số 31 , 0 1 10, 0 x khi x gx x x khi x tại 0 x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 10. Tìm m để hàm số 3 22 1 , 1 1 , 1 x khi x hx x mx x m khi x có giới hạn tại 1 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài 11. Cho hàm số 2 2 32 khi 1 1 khi 1 2 xx x x fx x x . Tính các giới hạn sau a). 1 lim x fx . b). 1 lim x fx . c). 1 lim x fx , (nếu có). Lời giải Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 1 2 3 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài 12. Cho hàm số 3 2 khi 1 khi 3 1 2 x fx x x x . Tìm 1 lim x fx . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 3. Câu hỏi trắc nghiệm. Câu 19. Cho hàm số 2 1 . 2 1 3 1 1 x x x fx xx ví i ví i Khi đó 1 lim x fx là: A. . B. 2. C. 4. D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 20. Cho hàm số 2 . 1 1 1 2 2 1 x x fx x xx ví i ví i Khi đó 1 lim x fx là: A. . B. 1. C. 0. D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 21. Cho hàm số 2 3 2 1 2 . xx fx xx víi víi Khi đó 2 lim x fx là: A. 1. B. 0. C. 1. D. Không tồn tại. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV-Bài 3. Giới Hạn Một Bên 1 2 4 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 22. Cho hàm số 2 3 2 1 . 2 xx fx ax x v í i v í i Tìm a để tồn tại 2 lim . x fx A. 1. a B. 2. a C. 3. a D. 4. a Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 23. Cho hàm số 2 2 2 3 3 1 . 3 3 2 3 x x x f x x xx víi víi víi Khẳng định nào dưới đây sai? A. 3 lim 6. x fx B. Không tồn tại 3 lim . x fx C. 3 lim 6. x fx D. 3 lim 15. x fx Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 24.(THTT Số 4 2018) Cho hàm số 42 khi 0 1 khi 0 4 x x x fx mx m x , m là tham số . Tìm giá trị của m để hàm số có giới hạn tại 0 x . A. 1 m . B. 0 m . C. 1 2 m . D. 1 2 m . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 2 5 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 A. L THU T I. Hàm số liên tục tại 1 đi ểm: 1. Đ ịnh nghĩa: Giả sử hàm số fx xác đ ịnh trên khoảng ; ab và 0 ;. x a b Hàm số y f x gọi là liên tục tại đi ểm 0 x nếu: 0 0 . lim xx f x f x Hàm số không liên tục tại đi ểm 0 x gọi là gián đo ạn tại 0 x . 2. Nhận xét. Để xét tính liên tục tại một đi ểm 0 x ta tiến hành hai bư ớc sau. Bước 1: Tính 0 fx . Bước 2: Tính 0 lim xx fx . Nếu 0 0 lim xx f x f x thì hàm số fx liên tục tại 0 x . 3. Ví dụ minh họa. Ví dụ 1 . Xét tính liên tục của các hàm số sau tại điểm 2 x a). 2 4 2 x fx x b). 2 4 2 2 4 2 x x gx x x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Ví dụ 2 . Cho hàm số: 2 2 35 2 4 1 2 6 x x x y f x x a). Tính 2 lim . x fx b). Xét tính liên tục của hàm số fx tại 2; x Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... BÀI 4. HÀM SỐ LIÊN TỤC Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 2 6 II. Hàm số liên tục trên một khoảng, trên một đo ạn. 1. Định nghĩa: Giả sử hàm số f xác đ ịnh trên khoảng ; ab .Ta nói rằng hàm số y f x liên tục trên khoảng ; ab nếu nó liên tục tại mọi đi ểm của khoảng đó. Hàm số y f x gọi là liên tục trên đo ạn ; ab nếu nó liên tục trên khoảng ; ab và lim lim xa xb f x f a f x f b 2. Các tính chất của hàm số liên tục. Định lí 1. Hàm số đa th ức liên tục trên . Hàm số phân thức, các hàm số lượng giác liên tục trên từng khoảng xác đ ịnh của chúng. Định lí 2. Giả sử y f x , y g x liên tục tại đi ểm 0 x . Khi đó Các hàm số , , . y f x g x y f x g x y f x g x liên tục tại 0 x Hàm số fx y gx liên tục tại 0 x nếu 0 0 gx . 3. Ví dụ minh họa. Ví dụ 3 . Chứng minh các hàm số sau liên tục trên . a). 42 2 f x x x b). 22 .sin 2cos 3 f x x x x c). 3 3 23 1 1 7 1 3 xx x x fx x d). 2 43 1 1 5 1 xx x fx x xx Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 2 7 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... III. Chứng minh phương trình có nghi ệm: 1. Định lí 2 (định lí về giá trị trung gian của hàm số liên tục) Giả sử hàm số f liên tục trên đo ạn ; ab . Nếu f a f b thì với mỗi số thực M nằm giữa , f a f b , tồn tại ít nhất một đi ểm ; c a b sao cho . f c M 2. nghĩa hình h ọc của đ ịnh lí Nếu hàm số f liên tục trên đo ạn ; ab và M là một số thực nằm giữa , f a f b thì đư ờng thẳng yM cắt đ ồ thị của hàm số y f x tại ít nhất một đi ểm có hoành đ ộ ; c a b . 3. Hệ quả Nếu hàm số f liên tục trên đo ạn ; ab và .0 f a f b thì tồn tại ít nhất một đi ểm ; c a b sao cho 0. fc 4. nghĩa hình h ọc của hệ quả Nếu hàm số f liên tục trên đo ạn ; ab và .0 f a f b thì đ ồ thị của hàm số y f x cắt trục hoành ít nhất tại một đi ểm có hoành đ ộ ; c a b . 5. Ví dụ minh họa. Ví dụ 4 . Chứng minh rằng phương trình 32 4 8 1 0 xx có nghiệm trong khoảng 1;2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Ví dụ 5 . Chứng minh phương trình 42 4 2 3 0 x x x có ít nhất 2 nghiệm thuộc 1;1 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 2 8 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... B. PHÂN DẠNG, PHƯƠNG PHÁP VÀ VÍ D Ụ MINH HỌA. DẠNG 1: Xét tính liên tục của hàm số tại một điểm. Ta xét dạng này gồm hai bài toán tổng quát sau. Bài toán 1. Cho hàm số 10 20 khi khi f x x x fx f x x x . Để xét tính liên tục hoặc xác định giá trị của tham số để hàm số liên tục tại điểm 0 x , ta thực hiện các bước sau 1. Phương pháp. Bước 1. Tính giới hạn 00 1 lim lim x x x x f x f x L . Bước 2. Tính 0 2 0 f x f x . Bước 3. Đánh giá ho ặc giải phương trình 20 L f x , từ đó đưa ra k ết luận. Nếu 20 L f x thì kết luận hàm số liên tục tại 0 x . Nếu 20 L f x thì kết luận hàm số không liên tục tại 0 x hay bị gián đo ạn tại 0 x . 2. Bài tập minh họa. Bài tập 1. Xét tính liên tục của hàm số 2 2 khi 2 2 2 2 khi 2 x x fx x x tại 2 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 2 . Xét tính liên tục của hàm số 2 2 4 khi 2 2 2 khi 2 x x fx xx x tại 2 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 2 9 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Bài tập 3 . Xét tính liên tục của hàm số 1 2 3 khi 2 2 1 khi 2 x x fx x x tại 2 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 4. Xét tính liên tục của hàm số sin khi 1 1 khi 1 x x fx x x tại 1 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 5. Tìm m để hàm số 32 22 khi 1 1 3 khi 1 x x x x fx x x m x liên tục tại 1 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 6. Tìm m để hàm số liên tục tại điểm đã chỉ ra 2 1 sin khi 0 khi 0 xx fx x mx tại 0 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 0 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 7. Tìm m để hàm số liên tục tại điểm đã chỉ ra 2 1 cos khi khi x x x fx mx tại x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 8. Tìm m để hàm số 2 22 2 khi 1 1 11 khi 1 xx mx x x fx xx x x liên tục tại 1 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài toán 2. Cho hàm số 10 20 khi khi f x x x fx f x x x . Để xét tính liên tục hoặc xác đ ịnh giá trị của tham số mđể hàm số liên tục tại đi ểm 0 x , ta làm các bư ớc 1. Phương pháp. Bước 1. Tính 0 2 0 f x f x . Bước 2. (Liên tục trái) Tính giới hạn 00 11 lim lim x x x x f x f x L . Đánh giá ho ặc giải phương trình 1 2 0 L f x , từ đó đưa ra k ết luận. Bước 3. (Liên tục phải) Tính giới hạn 00 12 lim lim x x x x f x f x L . Đánh giá ho ặc giải phương trình 2 2 0 L f x , từ đó đưa ra k ết luận. Bước 4. Đánh giá ho ặc giải phương trình 12 LL , từ đó đưa ra k ết luận. Nếu 1 2 2 0 L L f x thì kết luận hàm số liên tục tại 0 x . Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 1 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Nếu 1 2 0 L f x hoặc 2 2 0 L f x hoặc 21 LL kết luận hàm số không liên tục tại 0 x hay bị gián đo ạn tại 0 x . 2. Bài tập minh họa. Bài tập 9. Xét tính liên tục của hàm số 2 5 khi 5 2 1 3 5 3 khi 5 x x x fx xx tại 5 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 10. Xét tính liên tục của hàm số 1 cos khi 0 1 khi 0 xx fx xx tại 0 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 11. Xét tính liên tục của hàm số 3 3 khi 0 2 11 khi 0 11 xx fx x x x tại 0 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 2 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 12. Tìm m để hàm số liên tục tại điểm đã chỉ ra 2 khi 1 2 khi 1 1 khi 1 x x x f x x mx x tại 1 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 13. Cho hàm số 2 32 khi 1 1 khi 1 xx x x fx ax . a). Tìm a để hàm số liên tục trái tại điểm 1 x . b). Tìm a để hàm số liên tục phải tại điểm 1 x . c). Tìm a để hàm số liên tục tại điểm 1 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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.................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 4. Câu hỏi trắc nghiệm. Câu 1. Hàm số 1 3 4 f x x x liên tục trên: A. 4;3 . B. 4;3 . C. 4;3 . D. ; 4 3; . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 3 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 2. Hàm số 3 cos sin 2sin 3 x x x x fx x liên tục trên: A. 1;1 . B. 1;5 . C. 3 ;. 2 D. . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 3. Cho hàm số fx xác định và liên tục trên với 2 32 1 xx fx x với mọi 1. x Tính 1. f A. 2. B. 1. C. 0. D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 4. Cho hàm số fx xác định và liên tục trên m với 33 xx fx x với 0 x . Tính 0 f . A. 23 . 3 B. 3 . 3 C. 1. D. 0. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 5. Cho hàm số fx xác định và liên tục trên 4; với 42 x fx x với 0 x . Tính 0 f . A. 0. B. 2. C. 4. D. 1. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 4 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 6. Tìm giá trị thực của tham số m để hàm số 2 2 khi 2 2 khi 2 xx x fx x mx liên tục tại 2. x A. 0. m B. 1. m C. 2. m D. 3. m Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 7. Tìm giá trị thực của tham số m để hàm số 32 22 khi 1 1 3 khi 1 x x x x fx x x m x liên tục tại 1. x A. 0. m B. 2. m C. 4. m D. 6. m Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 8. Tìm giá trị thực của tham số k để hàm số 1 khi 1 1 1 khi 1 x x y f x x kx liên tục tại 1. x A. 1 . 2 k B. 2. k C. 1 . 2 k D. 0. k Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 9. Biết rằng hàm số 3 khi 3 12 khi 3 x x fx x mx liên tục tại 3 x (với m là tham số). Khẳng định nào dưới đây đúng? A. 3;0 . m B. 3. m C. 0;5 . m D. 5; . m Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 5 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 10. Tìm giá trị thực của tham số m để hàm số 2 1 sin khi 0 khi 0 xx fx x mx liên tục tại 0. x A. 2; 1 . m B. 2. m C. 1;7 . m D. 7; . m Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 11. Biết rằng 0 sin lim 1. x x x Hàm số tan khi 0 0 khi 0 x x fx x x liên tục trên khoảng nào sau đây ? A. 0; . 2 B. 0 x C. ;. 44 D. ;. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 12. Biết rằng 0 sin lim 1. x x x Tìm giá trị thực của tham số m để hàm số sin khi 1 1 khi 1 x x fx x mx liên tục tại 1. x A. . m B. . m C. 1. m D. 1. m Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 6 Câu 13. Biết rằng 0 sin lim 1. x x x Tìm giá trị thực của tham số m để hàm số 2 1 cos khi khi x x x fx mx liên tục tại . x A. . 2 m B. . 2 m C. 1 . 2 m D. 1 . 2 m Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 14. Hàm số 4 2 3 khi 1 khi 1, 0 1 khi 0 x xx f x x x xx x liên tục tại: A. mọi điểm trừ 0, 1. xx B. mọi điểm . x C. mọi điểm trừ 1. x D. mọi điểm trừ 0. x Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Câu 15. Số điểm gián đoạn của hàm số 2 0,5 khi 1 1 khi 1, 1 1 1 khi 1 x xx f x x x x x là: A. 0. B. 1. C. 2. D. 3. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 7 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... DẠNG 2: Xét tính liên tục của hàm số trên . Ta xét dạng này gồm bài toán tổng quát sau. Bài toán 1. Cho hàm số 10 20 khi khi f x x x fx f x x x . Để xét tính liên tục hoặc xác đ ịnh giá trị của tham số để hàm số liên tục tại đi ểm 0 x , ta làm các bư ớc 1. Phương pháp. Bước 1. Xét trên khoảng 0 xx . Nếu là đa th ức, hàm lư ợng giác, hàm phân thức hữu tỉ thì kết luận liên tục trên khoảng 0 ;. x Bước 2. Xét trên khoảng 0 xx . Nếu là đa th ức, hàm lư ợng giác, hàm phân thức hữu tỉ thì kết luận liên tục trên khoảng 0 ;. x Bước 3. Xét liên tục tại 0 xx . Đưa v ề dạng toán 1: xét tinh liên tục tại đi ểm 0 xx gồm hai bài toán 1 và 2. Bước 4. Kết luận. 2. Bài tập minh họa. Bài tập 14. Cho hàm số fx xác định bởi 2 2 khi 3 3 khi 1 3 12 x x x fx x x x . Chứng minh rằng hàm số liên tục trên khoảng 1; . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 8 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 15. Xác định a để hàm số 2 1 khi 1 1 khi 1 x x fx x ax liên tục trên đoạn 0;1 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 16. Cho hàm số 1 3 3 5 7 5 x f x ax b x x . Xác định a, b để hàm số liên tục trên R. Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 3 9 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Bài tập 17. Xét xem các hàm số sau có liên tục với xR không? Nếu không? Chỉ ra các điểm gián đoạn. a). 43 2 4 2 1 f x x x x b). 2 2 3 4 5 32 xx fx xx c). 2 2 1 1 2 3 1 2 1 2 2 x x xx fx x d). 32 2 6 3 3 3 19 3 x x x x fx x x Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 18. Cho hàm số 3 2 8 2 4 3 2 3 5 3 2 x khi x x f x khi x x khi x . Tìm các khoảng, nửa khoảng mà trên đó hàm số f(x) liên tục. Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 0 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 19. Cho hàm số 2 32 2 2 2 xx x fx x ax Với giá trị nào của a thì hàm số đã cho liên tục tại điểm 2 x ? Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 20. Cho hàm số 2 2 7 6 khi x < 2 2 1 a + 2 2 xx x y f x x khi x x . Xác định a để hàm số f(x) liên tục tại 0 2 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 1 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 3. Câu hỏi trắc nghiệm . Câu 16. Có bao nhiêu giá trị thực của m để hàm số 22 khi 2 1 khi 2 m x x fx m x x liên tục trên ? A. 2 . B. 1. C. 0 . D. 3 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 17. Biết rằng hàm số 0;4 4;6 khi 1 khi xx fx mx tục trên 0;6 . Khẳng định nào sau đây đúng? A. 2. m B. 2 3. m C. 3 5. m D. 5. m Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Câu 18. Có bao nhiêu giá trị tham số a để hàm số 2 32 khi 1 1 khi 1 xx x x fx ax liên tục trên . A. 1. B. 2 . C. 0 . D. 3 . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 2 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 19. Biết rằng 2 1 khi 1 1 khi 1 x x fx x ax liên tục trên đoạn 0;1 (với a là tham số). Khẳng định nào dưới đây về giá trị a là đúng? A. a là một số nguyên. B. a là một số vô tỉ. C. 5. a D. 0. a Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 20. Xét tính liên tục của hàm số . 1 khi 1 21 2 khi 1 x x fx x xx Khẳng định nào dưới đây đúng? A. fx không liên tục trên . B. fx không liên tục trên 0;2 . C. fx gián đoạn tại 1. x D. fx liên tục trên . 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Câu 21. Tìm giá trị nhỏ nhất của a để hàm số 2 2 56 khi 3 43 1 khi 3 xx x fx xx a x x liên tục tại 3 x . A. 2 3 . B. 2 . 3 C. 4 . 3 D. 4 . 3 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 3 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 22. Tìm giá trị lớn nhất của a để hàm số 3 2 3 2 2 khi 2 2 1 khi 2 4 x x x fx a x x liên tục tại 2. x A. max 3. a B. max 0. a C. max 1. a D. max 2. a Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 23. Xét tính liên tục của hàm số 1 cos khi 0 1 . khi 0 xx x fx x Khẳng định nào sau đây đúng? A. fx liên tục tại 0. x B. fx liên tục trên ;1 . C. fx không liên tục trên . D. fx gián đoạn tại 1. x Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 24. Tìm các khoảng liên tục của hàm số cos khi 1 2 1 . khi 1 x x x fx x Mệnh đề nào sau là sai? A. Hàm số liên tục tại 1 x . B. Hàm số liên tục trên các khoảng . ; , 1 1; C. Hàm số liên tục tại 1 x . D. Hàm số liên tục trên khoảng 1,1 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 4 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 25. Hàm số fx có đồ thị như hình bên không liên tục tại điểm có hoành độ là bao nhiêu? A. 0. x B. 1. x C. 2. x D. 3. x Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 26. Cho hàm số 2 khi 1, 0 0 khi 0 . khi 1 x xx x f x x xx Hàm số fx liên tục tại: A. mọi điểm thuộc . B. mọi điểm trừ 0 x . C. mọi điểm trừ 1 x . D. mọi điểm trừ 0 x và 1 x . 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Câu 27. Cho hàm số 2 1 khi 3, 1 1 4 khi 1 1 khi 3 x xx x f x x xx . Hàm số fx liên tục tại: A. Mọi điểm thuộc . B. Mọi điểm trừ 1 x . C. Mọi điểm trừ 3 x . D. Mọi điểm trừ 1 x và 3 x . Lời giải. Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 5 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 28. Số điểm gián đoạn của hàm số 2 2 khi 0 1 khi 0 2 3 1 khi 2 xx h x x x xx là: A. 1. B. 2. C. 3. D. 0. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 29. Tính tổng S gồm tất cả các giá trị m để hàm số 2 2 khi 1 2 khi 1 1 khi 1 x x x f x x m x x liên tục tại 1 x A. 1. S B. 0. S C. 1. S D. 2. S Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 30. Cho hàm số 2 3 cos khi 0 khi 0 1. 1 khi 1 x x x x f x x x xx Hàm số fx liên tục tại: A. mọi điểm thuộc . x B. mọi điểm trừ 0. x C. mọi điểm trừ 1. x D. mọi điểm trừ 0; 1. xx Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 6 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Dạng 3: Chứng minh phương trình có nghi ệm. Ta xét bài toán tổng quát sau. Bài toán 1. Cho phương trình 0 fx . Chứng minh phương trình có nghiệm. 1. Phương pháp. Bước 1: Biến đ ổi phương trình v ề dạng 0 fx . Bước 2: Hàm số liên tục và xác đ ịnh trên tập xác đ ịnh hàm số liên tục trên ;. ab Bước 3: Tìm hai số a và b sao cho .0 f a f b . Từ đó suy ra phương trình 0 fx có ít nhất một nghiệm thuộc ; ab . Chú ý: Nếu .0 f a f b thì phương trình có ít nh ất một nghiệm thuộc ; ab Nếu hàm số fx liên tục trên ; a và có . lim 0 x f a f x thì phương trình 0 fx có ít nhất một nghiệm thuộc ; a . Nếu hàm số f(x) liên tục trên ;a và có . lim 0 x f a f x thì phương trình 0 fx có ít nhất một nghiệm thuộc ;a . Để chứng minh 0 fx có ít nhất n nghiệm trên ; ab , ta chia đo ạn ; ab thành n đoạn nhỏ rời nhau, rồi chứng minh trên mỗi khoảng đó phương trình có ít nh ất một nghiệm. 2. Bài tập minh họa. Bài tập 21. Chứng minh rằng phương trình a). 42 3 5 6 0 x x x có ít nhất một nghiệm thuộc khoảng 1;2 . b). 3 10 xx có ít nhất một nghiệm âm lớn hơn 1 . c). 2 cos sin 1 0 x x x x có ít nhất một nghiệm thuộc khoảng 0; . 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 7 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 22. Chứng minh phương trình 53 5 4 1 0 x x x có đúng năm nghiệm. Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 23. Chứng minh rằng các phương trình sau luôn có nghiệm a). 5 3 3 0 xx . b). 4 3 2 3 1 0 x x x x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 8 Bài tập 24. Chứng minh rằng phương trình a). 42 4 2 3 0 x x x có ít nhất hai nghiệm phân biệt thuộc khoảng 1;1 . b). 54 5 4 1 0 x x x có ít nhất ba nghiệm phân biệt thuộc khoảng 0;5 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 25. Chứng minh rằng phương trình a). 2 2 3 4 0 xx có hai nghiệm phân biệt thuộc khoảng 3;1 . b). 32 3 3 0 xx có ba nghiệm phân biệt thuộc khoảng 1;3 . c). 3 2 6 1 3 xx có ba nghiệm phân biệt thuộc khoảng 7;9 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 4 9 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài toán 2. Chứng minh phương trình có ch ứa tham số m luôn có nghiệm với mọi . m 1. Phương pháp. Bước 1. Chọn hai số a và b thỏa mãn hai trư ờng hợp. 0 fa thì 0, f b m (đưa v ề bình phương thi ếu khi biểu thức có chứa tham số ) m . 0, f a f b m nên phương trình luôn có ít nh ất một nghiệm . m 0 fa thì 0, f b m (đưa v ề bình phương thi ếu khi biểu thức có chứa tham số ) m . 0, f a f b m nên phương trình luôn có ít nh ất một nghiệm . m Bước 2. Kết luận 2. Bài tập minh họa. Bài tập 25. Chứng minh rằng các phương trình sau luôn có nghiệm a). 3 22 1 1 3 0 m x x x . b). 2 7 5 5 1 0 m m x x . c). 43 1 2 1 3 0 m x x x x . d). cos cos2 0 x m x e). 2cos 2 2sin5 1 m x x . f). 11 cos sin m xx Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 5 0 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 25. Chứng minh rằng nếu 3 m thì phương trình 2 3 2 3 1 3 2 1 3 0 m m x m x m x có ít nhất một nghiệm thuộc khoảng 1;1 . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài toán 2. Chứng minh phương trình có ch ứa tham số m luôn có nghiệm dương ho ặc nghiệm âm với mọi . m 1. Phương pháp. Chọn số a thỏa mãn hai trư ờng hợp. Trường hợp 1. Nếu nghiệm dương ch ọn 0 a Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 5 1 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 Khi đó hàm số fx liên tục trên ; a thỏa 00 lim 0 x tính f f x . lim 0 x f a f x thì phương trình 0 fx có ít nhất một nghiệm dương thuộc ; a . Trường hợp 2. Nếu nghiệm âm chọn 0 a Thì hàm số fx liên tục trên ;a thỏa 00 lim 0 x tính f f x . lim 0 x f a f x thì phương trình 0 fx có ít nhất một nghiệm âm thuộc ;a . 2. Bài tập minh họa. Bài tập 25. Chứng minh rằng phương trình a). 32 10 x mx luôn có một nghiệm dương. b). 3 11 x mx m luôn có một nghiệm lớn hơn 1. c). 2 2 * 3 2 4 0, n m m x x n luôn có ít nhất một nghiệm âm với mọi . m Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 25. Tìm m để phương trình 32 3 2 2 3 0 x x m x m có ba nghiệm phân biệt 1 2 3 , , x x x thỏa mãn 1 2 3 1 x x x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 5 2 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 25. Chứng minh phương trình 4 30 xx luôn có ít nhất một nghiệm 0 x thỏa mãn điều kiện 7 0 12 x . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Bài tập 25. Cho , , a b c là các số thực khác 0. Chứng minh rằng các phương trình sau luôn có nghiệm a). 2 0 ax bx c với 2 3 6 0. a b c b). 2 0 ax bx c với 0 21 a b c m m m và 0 m . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 5 3 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 25. Chứng minh rằng nếu 2 3 6 0 a b c thì phương trình 2 tan tan 0 a x b x c có ít nhất một nghiệm trong khoảng ; 4 kk . Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Bài tập 25. Chứng minh rằng mọi phương trình bậc ba 32 0 ax bx cx d 0 a luôn có ít nhất một nghiệm. 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Bài tập 25. Chứng minh rằng mọi phương trình bậc bốn 4 3 2 0 ax bx cx dx e với .0 ae luôn có ít nhất một nghiệm. Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 5 4 Bài tập 25. Cho , , a b c là ba số dương phân biệt. Chứng minh rằng phương trình 0 a x b x c b x a x c c x a x b luôn có hai nghiệm phận biệt. Lời giải .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 4. Câu hỏi trắc nghiệm . Câu 31. Cho hàm số 3 4 4 1. f x x x Mệnh đề nào sau đây là sai? A. Hàm số đã cho liên tục trên . B. Phương trình 0 fx không có nghiệm trên khoảng ;1 . C. Phương trình 0 fx có nghiệm trên khoả ng 2;0 . D. Phương trình 0 fx có ít nhất hai nghiệm trên khoảng 1 3; . 2 Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... 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Câu 32. Cho phương trình 42 2 5 1 0. x x x Mệnh đề nào sau đây là đúng? A. Phương trình không có nghiệm trong khoảng 1;1 . B. Phương trình không có nghiệm trong khoảng 2;0 . C. Phương trình chỉ có một nghiệm trong khoảng 2;1 . D. Phương trình có ít nhất hai nghiệm trong khoảng 0;2 . Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. Hàm Số Liên Tục 1 5 5 Lớp Toán Thầy – Diệp Tuân Tel: 0935.660.880 .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 33. Cho hàm số 3 31 () f x x x . Số nghiệm của phương trình 0 fx trên là: A. 0. B. 1. C. 2. D. 3. 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Câu 34. Cho hàm số fx liên tục trên đoạn 1;4 sao cho 12 f , 47 f . Có thể nói gì về số nghiệm của phương trình 5 fx trên đoạn [ 1;4] : A. Vô nghiệm. B. Có ít nhất một nghiệm. C. Có đúng một nghiệm. D. Có đúng hai nghiệm. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Câu 35. Có tất cả bao nhiêu giá trị nguyên của tham số m thuộc khoảng 10;10 để phương trình 32 3 2 2 3 0 x x m x m có ba nghiệm phân biệt 1 2 3 , , x x x thỏa mãn 1 2 3 1 x x x ? A. 19. B. 18. C. 4. D. 3. Lời giải. .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... .................................................................................................................................................................................................................... Trung Tâm Luyện Thi Đại Học Amsterdam Chương IV -Bài 4. 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