Although it is a very simple question, I am not able to get similar results using Stirling's approximation as obtained using Integration. Here is what I have attempted.
$$L=\lim_{n\to\infty} \left(\frac{n!}{n^n}\right)^{1/n}$$
$$\begin{aligned}L&=\lim_{n\to\infty}\left(\frac{\sqrt{2\pi n}}{e^{n}}\right)^{1/n}\\&=\lim_{n\to\infty}\frac{1}{e}n^{1/2n}\end{aligned}$$
I am not able to reason out as to how this will evaluate to $1/e$. Any hints are appreciated. Thanks.