Đáp án:1)\(\frac{1}{4}\)
2)\(\frac{1}{4}\)
3)\(-\infty \)
4)0
5)\(\frac{1}{4}\)
6)\(\frac{1}{60}\)
7)\(\frac{2}{3}\)
8)\(\frac{-1}{2}\)
Giải thích các bước giải:
1)\(\lim_{x\rightarrow 0}\frac{\sqrt{x+4}-2}{x}\)
=\(\lim_{x\rightarrow 0}\frac{x+4-4}{x(\sqrt{x+4}+2)}\)
=\(\lim_{x\rightarrow 0}\frac{1}{\sqrt{x+4}+2}\)
=\(\frac{1}{4}\)
2)\(\lim_{x\rightarrow 1}\frac{\sqrt{x+3}-2}{x-1}\)
=\(\lim_{x\rightarrow 1}\frac{x+3-4}{(x-1)(\sqrt{x+3}+2)}\)
=\(\lim_{x\rightarrow 1}\frac{x-1}{(x-1)\sqrt{x+4}+2}\)
=\(\lim_{x\rightarrow 1}\frac{1}{\sqrt{x+4}+2}\)
=\(\frac{1}{4}\)
3)\(\lim_{x\rightarrow 7}\frac{2-\sqrt{x-2}}{x^{2}-49}\)
=\(\lim_{x\rightarrow 7}\frac{4-x+2}{(x^{2}-49)(2+\sqrt{x-2}}\)
=\(\lim_{x\rightarrow 7}\frac{-x+6}{(x^{2}-49)(2+\sqrt{x-2}}\)
=\(\lim_{x\rightarrow 7}\frac{-1}{0}\)
=\(-\infty \)
4)\(\lim_{x\rightarrow 1}\frac{x-\sqrt{2x-1}}{x^{2}-12x+11}\)
=\(\lim_{x\rightarrow 1}\frac{x^{2}-2x+1}{(x-1)(x-11)(x+\sqrt{2x-1})}\)
=\(\lim_{x\rightarrow 1}\frac{(x-1)^{2}}{(x-1)(x-11)(x+\sqrt{2x-1})}\)
=\(\lim_{x\rightarrow 1}\frac{x-1}{(x-11)(x+\sqrt{2x-1})}\)
=0
5)\(\lim_{x\rightarrow 6}\frac{\sqrt{x-2}-2}{x-6}\)
=\(\frac{x-2-4}{(x-6)(\sqrt{x-2}+2)}\)
=\(\frac{x-6}{(x-6)(\sqrt{x-2}+2)}\)
=\(\frac{1}{\sqrt{x-2}+2}\)
=\(\frac{1}{4}\)
6)\(\lim_{x\rightarrow 5}\frac{\sqrt{x+4}-3}{x^{2}-25}\)
=\(\lim_{x\rightarrow 5}\frac{x+4-9}{(x-5)(x+5)(\sqrt{x+4}+3)}\)
=\(\lim_{x\rightarrow 5}\frac{x-5}{(x-5)(x+5)(\sqrt{x+4}+3)}\)
=\(\lim_{x\rightarrow 5}\frac{1}{(x+5)(\sqrt{x+4}+3)}\)
=\(\frac{1}{60}\)
7)\(\lim_{x\rightarrow 2}\frac{\sqrt{x^{2}+5}-3}{x-2}\)
=\(\frac{x^{2}+5-9}{(x-2)(\sqrt{x^{2}+5}+3)}\)
=\(\frac{x^{2}-4}{(x-2)(\sqrt{x^{2}+5}+3)}\)
=\(\frac{(x-2)(x+2)}{(x-2)(\sqrt{x^{2}+5}+3)}\)
=\(\frac{x+2}{\sqrt{x^{2}+5}+3}\)
=\(\frac{2}{3}\)
8)\(\lim_{x\rightarrow 0}\frac{\sqrt{x^{3}+1}-1}{x^{2}+x}\)
=\(\lim_{x\rightarrow 0}\frac{(x+1)(x^{2}-x+1)-1}{x(x+1)(\sqrt{x^{3}+1}+1)}\)
=\(\lim_{x\rightarrow 0}\frac{x^{2}-x+1-1}{x(\sqrt{x^{3}+1}+1)}\)
=\(\lim_{x\rightarrow 0}\frac{x(x-1)}{x(\sqrt{x^{3}+1}+1)}\)
=\(\lim_{x\rightarrow 0}\frac{x-1}{\sqrt{x^{3}+1}+1}\)
=\(\frac{-1}{2}\)
