calculus - $lim_{n to infty} frac{sqrt[n]{n!}}{n}$

I tried to do a change of variable with $t=\frac{1}{n}$ in order to use Taylor expansion but the factor $n!$ becomes a fractional number (during the lesson it was define only for $\mathbb{Z}$ set number).
Another attempt I made was to write the limit as $\lim_{n \to \infty} \sqrt[n]{\frac{n!}{n^n}} \rightarrow e^{\frac{1}{n}log{\frac{n!}{n^n}}}$ but it doesn't work. After that I tried to use the Stirling formula and yes... it works! But I would find something that doesn't use this formula.
Some advice?