階乗の冪婚を含む極限値問題

\begin{align*} \lim_{n\rightarrow\infty}\frac{n}{\sqrt[n]{n!}} & =\lim_{n\rightarrow\infty}\exp\left(\log\frac{n}{\sqrt[n]{n!}}\right)\\ & =\lim_{n\rightarrow\infty}\exp\left(\log n-\frac{1}{n}\log n!\right)\\ & =\lim_{n\rightarrow\infty}\exp\left(-\frac{1}{n}\sum_{k=1}^{n}\log\frac{k}{n}\right)\\ & =\exp\left(-\int_{0}^{1}\log xdx\right)\\ & =\exp\left(-\left[x\log x-x\right]_{0}^{1}\right)\\ & =\exp\left(1\right)\\ & =e \end{align*}