How to evaluate $\lim _{x\to 1}\left(\frac{x+2}{x-1}-\frac{3}{\ln x}\right)$? I have a problem with this limit, I have no idea how to compute it. Can you explain the method and the steps used (without L'Hôpital if is possible)? Thanks
$$\lim _{x\to 1}\left(\frac{x+2}{x-1}-\frac{3}{\ln x}\right)$$

As $x \to 1$, one may use Taylor's series expansion to get
$$
\ln x= \ln (1+(x-1))=(x-1)-\frac12(x-1)^2+O((x-1)^3)
$$ or
$$
\frac{x+2}{x-1}-\frac{3}{\ln x}=\frac{x+2}{x-1}-\frac3{(x-1)-\frac12(x-1)^2+O((x-1)^3)}
$$ that is, as $x \to 1$, 
$$
\frac{x+2}{x-1}-\frac{3}{\ln x}=-\frac12+O(x-1)
$$ from which you deduce the sought limit.
