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Question Number 190501 by 073 last updated on 04/Apr/23

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{3^{\mathrm{x}}}{\mathrm{x}!}=?$$$$\mathrm{solution}?$$

Answered by Frix last updated on 04/Apr/23

$$0$$$$x!\sim \frac{x^x}{{\mathrm{e}}^x}\sqrt{2\pi x}$$$$\frac{n^x}{x!}\sim \frac{n^x{\mathrm{e}}^x}{x^x\sqrt{2\pi x}}={\left(\frac{3n}{x}\right)}^x\frac{1}{\sqrt{2\pi x}}\sim 0\mathrm{\forall }n\in {\mathbb{N}}^{★}$$

Commented by 073 last updated on 04/Apr/23

$$\mathrm{nice}\mathrm{solution}$$

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