Đáp án + giải thích các bước giải:
a) `lim_{x->0}(\sqrt{1+2x}-1)/(2x)=lim_{x->0}(1+2x-1)/(2x(\sqrt{1+2x}+1))=lim_{x->0}1/(\sqrt{1+2x}+1)=1/(1+1)=1/2 `
b) `lim_{x->2}(x-\sqrt{3x-2})/(x^2-4)=lim_{x->2}(x^2-(3x-2))/((x-2)(x+2)(x+\sqrt{3x-2}))=lim_{x->2}(x^2-3x+2)/((x-2)(x+2)(x+\sqrt{3x-2}))=lim_{x->2}((x-2)(x-1))/((x-2)(x+2)(x+\sqrt{3x-2}))=lim_{x->2}(x-1)/((x+2)(x+\sqrt{3x-2}))=(2-1)/((2+2)(2+\sqrt{3.2-2}))=1/16`
c) `lim_{x->0}(\sqrt{1+x^2}-1)/(2x^3-3x^2)=lim_{x->0}(1+x^2-1)/(x^2(2x-3)(\sqrt{1+x^2}+1))=lim_{x->0}1/((2x-3)(\sqrt{1+x^2}+1))=1/(-3.2)=-1/6`
d) `lim_{x->1}(\sqrt{2x+7}-x-2)/(x^3-4x+3)=lim_{x->1}(\sqrt{2x+7}-x-2)/(x^3-x^2+x^2-x-3x+3)=lim_{x->1}(\sqrt{2x+7}-x-2)/(x^2(x-1)+x(x-1)-3(x-1))=lim_{x->1}(2x+7-(x+2)^2)/((x^2+x-3)(x-1)(\sqrt{2x+7}+x+2))=lim_{x->1}(-(x^2+2x-3))/((x^2+x-3)(x-1)(\sqrt{2x+7}+x+2))=lim_{x->1}(-(x+3)(x-1))/((x^2+x-3)(x-1)(\sqrt{2x+7}+x+2))=lim_{x->1}(-(x+3))/((x^2+x-3)(\sqrt{2x+7}+x+2))=(-(1+3))/((1+1-3)(\sqrt{2+7}+1+2))=2/3`
