Question Number 190501 by 073 last updated on 04/Apr/23
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{3}^{\mathrm{x}} }{\mathrm{x}!}=? \\ $$$$\mathrm{solution}? \\ $$
Answered by Frix last updated on 04/Apr/23
$$\mathrm{0} \\ $$$${x}!\sim\frac{{x}^{{x}} }{\mathrm{e}^{{x}} }\sqrt{\mathrm{2}\pi{x}} \\ $$$$\frac{{n}^{{x}} }{{x}!}\sim\frac{{n}^{{x}} \mathrm{e}^{{x}} }{{x}^{{x}} \sqrt{\mathrm{2}\pi{x}}}=\left(\frac{\mathrm{3}{n}}{{x}}\right)^{{x}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{2}\pi{x}}}\sim\mathrm{0}\forall{n}\in\mathbb{N}^{\bigstar} \\ $$
Commented by 073 last updated on 04/Apr/23
$$\mathrm{nice}\:\mathrm{solution} \\ $$