I have this really nasty limit that I've been banging my head on all night. The question asks me to calculate the following:

$\displaystyle \lim_{n\to\infty}\frac{n \times \left(\frac{2n-2}{2n-1}\right)^{2n-2}}{2n-1}$

The limit evaluates to $\displaystyle \frac{\infty}{\infty}$, so my first instinct was to use L'Hospital's Rule and take the limit of $\displaystyle \frac{f'(x)}{g'(x)}$. But that didn't really simplify it; doing that just gave another nasty limit. Is there a simpler approach?

According to Wolfram Alpha, the limit is equal to $\displaystyle \frac{1}{2e}$. But it just says "series expansion at $\displaystyle n=\infty$" without showing any of the steps.

I know that $\displaystyle e$ is the sum of an infinite series. But how do you recognize that from the limit? And how do you get to $\displaystyle \frac{1}{2e}$?

Any pointers would be greatly appreciated!

- A really puzzled math student.