\[ \begin{align}
\lim_{n \to +\infty }\frac{\sqrt[n]{(n!)}}{n} &= \exp(\ln(\lim_{n \to +\infty }\frac{\sqrt[n]{(n!)}}{n})) \\
~ &= \exp(\lim_{n \to +\infty }\ln{\frac{\sqrt[n]{(n!)}}{n}}) \\
~ &= \exp(\lim_{n \to +\infty }\ln{\sqrt[n]{\frac{(n!)}{n^n} }}) \\
~ &= \exp(\lim_{n \to +\infty } \frac{1}{n} \ln{\frac{(n!)}{n^n} }) \\
~ &= \exp(\lim_{n \to +\infty } \frac{1}{n} \ln{(\frac{1}{n}\frac{2}{n}\cdots\frac{n}{n})}) \\
~ &= \exp(\lim_{n \to +\infty } \frac{1}{n} (\ln{(\frac{1}{n})} + \ln{(\frac{2}{n})} + \cdots + \ln{(\frac{n}{n})})) \\
~ &= \exp(\lim_{\epsilon \to 0^+ } \int^1_\epsilon {\ln(t)dt}) \\
~ &= \exp(\int^1_0 {\ln(t)dt}) \\
~ &= \exp(-1) \\
~ &= \frac{1}{e} \\
\end{align}\]