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Question Number 6028 by FilupSmith last updated on 10/Jun/16

Can you please show me how to solve :

L = \lim_{n \to \infty} \frac{x^{n}}{n!}

Commented by Yozzii last updated on 10/Jun/16

A s s u m e x > 0 .

n! \approx \sqrt{π (2 n + \frac{1}{3})} n^{n} e^{- n} f o r l a r g e n .

∴ F o r l a r g e n,

\frac{x^{n}}{n!} \approx \frac{{(e x)}^{n}}{\sqrt{π (2 n + \frac{1}{3})} n^{n}}

\frac{x^{n}}{n!} \approx \frac{1}{\sqrt{π (2 n + \frac{1}{3})}} {(\frac{e x}{n})}^{n}

Commented by Yozzii last updated on 10/Jun/16

L e t u = \frac{1}{n}

∴ \frac{x^{n}}{n!} \approx \frac{{(e x u)}^{1 / u}}{\sqrt{π (\frac{2}{u} + \frac{1}{3})}} u \to 0 a s n \to \infty

\lim_{n \to \infty} \frac{x^{n}}{n!} \approx \lim_{u \to 0} \frac{{(e x u)}^{1 / u} {(3 u)}^{0.5}}{\sqrt{π (6 + u)}}

\lim_{n \to \infty} \frac{x^{n}}{n!} \approx \lim_{u \to 0} \frac{{(e x)}^{1 / u} u^{\frac{1}{u} + 0.5} \sqrt{3}}{\sqrt{π (u + 6)}}

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