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Question Number 41855 by 123456780 last updated on 14/Aug/18

$$\underset{x\to +\mathrm{\infty }}{\lim }\frac{\mathrm{n}!}{\ln (1+\mathrm{n}!)}$$$$\mathrm{find}\mathrm{the}\mathrm{limit}$$

Commented by prof Abdo imad last updated on 14/Aug/18

$$letputp=n!\Rightarrow \frac{n!}{ln(1+n!)}=\frac{p}{ln(1+p)}$$$$=\frac{p}{lnp+ln(1+\frac{1}{p})}\sim \frac{p}{lnp+\frac{1}{p}}\sim \frac{p}{ln\left(p\right)}(p\to +\mathrm{\infty })$$$$but{lim}_{p\to +\mathrm{\infty }}\frac{p}{ln\left(p\right)}=+\mathrm{\infty }\Rightarrow$$$${lim}_{n\to +\mathrm{\infty }}\frac{n!}{ln(1+n!)}=+\mathrm{\infty }$$$$$$

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