lim-x-n-ln-1-n-find-the-limit-

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

\lim_{x \to + \infty} \frac{n!}{\ln (1 + n!)}

find the limit

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

l e t p u t p = n! \Rightarrow \frac{n!}{l n (1 + n!)} = \frac{p}{l n (1 + p)}

= \frac{p}{l n p + l n (1 + \frac{1}{p})} \sim \frac{p}{l n p + \frac{1}{p}} \sim \frac{p}{l n (p)} (p \to + \infty)

b u t {l i m}_{p \to + \infty} \frac{p}{l n (p)} = + \infty \Rightarrow

{l i m}_{n \to + \infty} \frac{n!}{l n (1 + n!)} = + \infty