Find the limit: $\lim \limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right)$ Find:
$$\lim\limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right)
$$
Without using L'Hospital or Taylor approximations
Thanks in advance

Let's write
$$\lim_{x\rightarrow1}\left({x\over x-1}-{1\over\ln x}\right)=1+\lim_{x\rightarrow1}\left({1\over x-1}-{1\over\ln x}\right)=1+L$$
and see if we can compute $L$.  But since $x\rightarrow1$ is equivalent to $x^2\rightarrow1$, we can rewrite $L$ as
$$L=\lim_{x\rightarrow1}\left({1\over x^2-1}-{1\over\ln x^2}\right)={1\over2}\lim_{x\rightarrow1}\left({-1\over x+1}+{1\over x-1}-{1\over\ln x}\right)={-1\over4}+{1\over2}L$$
So if we assume the limit exists, then we easily find $L=-1/2$, hence the desired limit is $1-1/2=1/2$.  Please note, though, this doesn't prove that the limit exists, just what its value is if it does.
