$\displaystyle \lim_{n->oo}\frac{(n!)^{1/n}}{n}$
It is commonly known that for convergent sequences the ration and root test gives the same result, in particular
$\displaystyle \lim_{n\to\infty}\frac{(n!)^{\frac{1}{n}}}{n}=\lim _{n\to\infty}\frac{(n+1)!}{(n+1)^{n+1}}\cdot\frac{ n^n}{n!}=\lim_{n\to\infty}\frac{n^n}{(n+1)^n}=\lim _{n\to\infty}\left(1+\frac{1}{n}\right)^n=e$