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Question Number 182659 by mathlove last updated on 12/Dec/22

$$\underset{n\to \mathrm{\infty }}{\lim }\frac{\sqrt[n]{n!}}{n}=?$$

Answered by Frix last updated on 12/Dec/22

$$n!\sim \frac{n^n\sqrt{2\pi n}}{{\mathrm{e}}^n}$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{\sqrt[n]{n!}}{n}=\underset{n\to \mathrm{\infty }}{\lim }\frac{{\left(2\pi n\right)}^{\frac{1}{2n}}}{\mathrm{e}}=\frac{1}{\mathrm{e}}$$

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