TITLE: How to find the limit $\lim\limits_{m\to\infty}\frac{m^{m-2}}{(m-1)^{m-2}}$?
QUESTION [9 upvotes]: I am trying to evaluate the limit of this:
$$\lim_{m\rightarrow \infty} \frac{m^{m-2}}{(m-1)^{m-2}}$$
That is just basic calculus I think but I forget those methods for finding the limit. I think the limit is just $e$. Anyone could prove that? Thanks!
Fei

REPLY [6 votes]: $$\lim_{m\to \infty} \frac{m^{m-2}}{(m-1)^{m-2}}=\lim_{m\to \infty}\left( \frac{m}{m-1}\right)^{m-2}=\lim_{m\to \infty}\left( \frac{m-1+1}{m-1}\right)^{m-2}=$$
$$=\lim_{m\to \infty}\left( 1+\frac{1}{m-1}\right)^{m-1-1}=\lim_{m\to \infty}\left( 1+\frac{1}{m-1}\right)^{m-1}\left( 1+\frac{1}{m-1}\right)^{-1}=$$
$$=\lim_{m\to \infty}\left( 1+\frac{1}{m-1}\right)^{m-1}\lim_{m\to \infty}\left( 1+\frac{1}{m-1}\right)^{-1}=e\cdot 1=e$$