What could this limit be?

Let If one replaces $r/n$ by $x$ and $1/n$ by $dx$ , the equation now becomes

I'm going to use Stirling formula $\displaystyle \lim_{n\to \infty} \left(\frac{n!}{n^n}\right)^{\frac{1}{n}} = \lim_{n\to \infty} \left(\frac{\sqrt{2\pi n}^{(\frac{1}{n})}\left(\frac{n}{e}\right)}{n}\right) = \frac{1}{e}$ due to $\displaystyle \lim_{n\to\infty} \sqrt[2n]{2\pi n} = 1$

using $epsilon$ - $delta$ definition :p