\displaystyle $\begin{align}
\ln \lim_{n \to \infty} \sqrt[n]{\frac{n^n}{n!}}
& = \lim_{n \to \infty} \ln \sqrt[n]{\frac{n^n}{n!}}
= \lim_{n \to \infty} \frac{n \ln n - \ln n!}{n} \\
& \!\!\! \stackrel{\sf Stolz}{=} \lim_{n \to \infty} \frac{(n + 1) \ln (n + 1) - \ln (n + 1)! - n \ln n + \ln n!}{(n + 1) - n} \\
& = \lim_{n \to \infty} \left( (n + 1) \ln (n + 1) - \ln (n + 1) - n \ln n \right)
= \lim_{n \to \infty} n (\ln (n + 1) - \ln n) \\
& = \lim_{n \to \infty} \ln \left( 1 + \frac{1}{n} \right)^n
= \ln \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n
= 1
\end{align}$