I need help proving this limit:
$$ \lim_{x \to \infty}\left[% {2{\rm Li}_{2}\left(1-{\rm e}^{x}\right) \over x} -{x\,{\rm e}^{x} \over 1 - {\rm e}^{x}}\right] =0 $$
Where the ${\rm Li}_{2}$ is polylogarithm function of order $2$.
The difficulty is because each component go to negative infinity, so we have to evaluate the limit of the whole expression. $$ \mbox{I was able to show that}\quad\frac{\displaystyle{\frac{x\,{\rm e}^{x}}{1 - {\rm e}^{x}}}} {\displaystyle{\frac{2\,{\rm Li}_{2}\left(1 - {\rm e}^{x}\right)}{x}}}=1 $$ but I have a hunch that the implication do not go that way. Please help quickly. My homework due soon. Thank you.