题目

求下列极限:lim _(xarrow 0)[ dfrac (1)(ln (1+x))-dfrac (1)(x)]

求下列极限:

$\lim_{x\rightarrow0}\left[\frac{1}{\ln\left(1+x\right)}-\frac{1}{x}^{ }\right]$

题目解答

答案

$\lim_{x\rightarrow0}\left(\frac{1}{\ln\left(1+x\right)}-\frac{1}{x}\right)=\lim_{x\rightarrow0}\frac{x-\ln\left(1+x\right)}{x\ln\left(1+x\right)}=\lim_{x\rightarrow0}\frac{x-\ln\left(1+x\right)}{x^2}=\lim_{x\rightarrow0}\frac{1-\frac{1}{1+x}}{2x}=\lim_{x\rightarrow0}\frac{\frac{1}{\left(1+x\right)^2}}{2}= \frac{1}{2}$