Question Number 172439 by Physicien last updated on 27/Jun/22
Commented by mr W last updated on 27/Jun/22
$${this}\:{app}\:{Equation}\:{Editor}\:{is}\:{made}\:{for} \\ $$$${writing}\:{equations}.\:{Why}\:{don}'{t}\:{you} \\ $$$${use}\:{it}\:{for}\:{writing}\:{equations}\:/\:{formulas} \\ $$$${instead}\:{of}\:{posting}\:{pictures}?\:{is}\:{it} \\ $$$${for}\:{you}\:{not}\:{easier}\:{and}\:{better}\:{to}\:{write} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}!}{{x}^{{x}} {e}^{−{x}} \sqrt{{x}}}\:?\:{to}\:{be}\:{honest},\:{the} \\ $$$${handwriting}\:{your}\:{pictures}\:{show}\:{is} \\ $$$${not}\:{the}\:{best}. \\ $$
Commented by Rasheed.Sindhi last updated on 27/Jun/22
$${BTW}\:{a}\:\boldsymbol{{typed}}\:\boldsymbol{{question}}\:{is}\:{much}\:{helpful} \\ $$$${in}\:{copy}-{paste}\:{also}. \\ $$
Answered by qaz last updated on 27/Jun/22
![lim_(n→∞) ((n!)/(n^n e^(−n) (√n)))=lim_(n→∞) (((√(2πn))((n/e))^n )/(n^n e^(−n) (√n)))=(√(2π)) [n!∼(√(2πn))((n/e))^n ,n→∞]](https://www.tinkutara.com/question/Q172446.png)
$$\underset{{n}\rightarrow\infty} {{lim}}\frac{{n}!}{{n}^{{n}} {e}^{−{n}} \sqrt{{n}}}=\underset{{n}\rightarrow\infty} {{lim}}\frac{\sqrt{\mathrm{2}\pi{n}}\left(\frac{{n}}{{e}}\right)^{{n}} }{{n}^{{n}} {e}^{−{n}} \sqrt{{n}}}=\sqrt{\mathrm{2}\pi}\:\:\:\:\:\:\left[{n}!\sim\sqrt{\mathrm{2}\pi{n}}\left(\frac{{n}}{{e}}\right)^{{n}} ,{n}\rightarrow\infty\right] \\ $$