Prove that $x^n$ is o(n!)

Question

Prove that for $ x \in \mathbb{R}$ $x^n$ is o(n!) I know that here we can use the definition of $e^{x}$ , But I would like to prove it with the limit definition, So $\lim_{n \rightarrow \infty}$ $\frac{x^n}{n!}$ would ratio test completely suffice here? Or are there other methods?

Answer 1

Take any integer $k >2|x|$. Note that $n! >(k)(k)... (k)=k^{n-k}$ if $n >k$. Hence $|\frac {x^{n}} {n!}|<k^{k} (\frac 1 2 )^{n} \to 0$