題目為：
$$
\lim _{n \rightarrow \infty} \frac{\sqrt[n]{n}}{n}=\frac{1}{e}
$$
解答：
$$
\begin{array}{c}
\ln \left(\frac{\sqrt[n]{n !}}{n}\right)=\frac{1}{n}(\ln 1+\ln 2+\cdots+\ln n)-\ln n \
=\frac{1}{n}\left(\ln \left(\frac{1}{n}\right)+\ln \left(\frac{2}{n}\right)+\cdots+\ln \left(\frac{n}{n}\right)\right) \longrightarrow \int_{0}^{1} \ln x d x=-1
\end{array}
$$