How to calculate $\lim_{n\to \infty } \frac{n^n}{n!^2}$? 
How to calculate $\lim_{n\to \infty } \frac{n^n}{n!^2}$?


I tried to solve this by using Stirling's formula but I got stuck at $\lim\limits_{n\to \infty } \frac{e^{2n}}{n^{n+1}}$. Any help?

Most of the time you do not need the exact Stirling approximation, just use the simplest: 
Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$
It ensues that $\dfrac{(n!)^2}{n^n}>\left(\dfrac n9\right)^n\to\infty$ so the inverse is going to $0$.

Anyway regarding the point where you are stuck $\dfrac{n^{n+1}}{e^{2n}}=\underbrace{n}_{\to\infty}\bigg(\ \underbrace{\dfrac n{e^2}}_{\to\infty}\ \bigg)^{\underbrace{n}_{\to\infty}}\to \infty$ 
so the inverse is going to $0$.
