limits - Compute $lim_{nrightarrowinfty}frac{n^n}{(n!)^2}$

I have to compute $\lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2}$.

I tried say that this limit exists and it's l, so we have $\lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2} = L$ then I rewrited it as:
$\lim_{n\rightarrow\infty}(\frac{\sqrt n}{\sqrt[n]{n!}})^{2n}$ then I used natural log over the whole expresion but didn't got into a nice place.

I don't know about Pi function or gamma function so therefore can't really use L'Hospital's rule.

Answer

By ratio test

$$\frac{(n+1)^{n+1}}{((n+1)!)^2}\frac{(n!)^2}{n^n}=\frac1{n+1}\left(1+\frac1n\right)^n\to 0$$

then

$$\lim_{n\rightarrow\infty}\frac{n^n}{(n!)^2}=0$$