$\LARGE {\color{DarkBlue} (1)\lim_{x\rightarrow 0}\frac{2^{x}-1}{x}=\lim_{x\rightarrow 0}\frac{2^{x}-1^{x}}{x}=ln2-ln1=ln2}$
[SIZE=6]${\color{DarkBlue} (2)\lim_{x\rightarrow 1}\frac{lnx}{x-1}=\lim_{x\rightarrow 1}ln(1+x-1)^{\frac{1}{x-1}}$
${\color{DarkBlue} =\lim_{(x-1)\rightarrow 0}ln(1+\frac{1}{\frac{1}{x-1}})^{\frac{1}{x-1}}}=lne=1}$ [/SIZE]
$\LARGE {\color{DarkBlue} (3)\lim_{x\rightarrow 0}\frac{ln(1+2x)}{x}=\lim_{x\rightarrow 0}ln(1+\frac{2}{\frac{1}{x}})^{\frac{1}{x}}=lne^{2 }=2}$
$\LARGE {\color{DarkBlue} (4)\lim_{x\rightarrow \infty }\frac{e^{-x}-1}{x}=\lim_{x\rightarrow \infty }\frac{\frac{1}{e^{x}}-1}{x}=0}$
${\color{DarkBlue} ()\lim_{x\rightarrow 0}\frac{a^{2x}-1}{x}=\lim_{x\rightarrow 0}\frac{(a^{2})^{x}-1^{x}}{x}=ln a^{2}}$
${\color{DarkBlue} \lim_{x\rightarrow 0}(1+3tan^{2}x)^{cot^{2}x}=\lim_{cotx\rightarrow \infty }(1+\frac{3}{cot^{2}x})^{cot^{2}x}=e^{3}}$
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