calculus - How to prove that $limlimits_{ntoinfty} frac{n!}{n^2}$ diverges to infinity?

$\lim\limits_{n\to\infty} \dfrac{n!}{n^2} \rightarrow \lim\limits_{n\to\infty}\dfrac{\left(n-1\right)!}{n}$

I can understand that this will go to infinity because the numerator grows faster.

I am trying to apply L'Hôpital's rule to this; however, have not been able to figure out how to take the derivative of $\left(n-1\right)!$

So how does one take the derivative of a factorial?

Answer

you could introduce the gamma function!

Just a joke, as $n!>n^3$ for $n>100$ you know that

$$\frac{n!}{n^2} > \frac{n^3}{n^2}=n$$