lim-x-3-n-n-2-n-n-

Question Number 136701 by mathlove last updated on 25/Mar/21

\lim_{x \to \infty} \frac{3^{n} - n!}{2^{n} - n!} = ?

Commented by yutytfjh67ihd last updated on 25/Mar/21

Commented by mathlove last updated on 25/Mar/21

h e l p m e

Answered by mathmax by abdo last updated on 25/Mar/21

\lim_{n \to + \infty} \frac{n! - 3^{n}}{n! - 2^{n}} = \lim_{n \to + \infty} \frac{1 - \frac{3^{n}}{n!}}{1 - \frac{2^{n}}{n!}}

n! \sim n^{n} e^{- n} \sqrt{2 π n} \Rightarrow \frac{3^{n}}{n!} \sim \frac{3^{n}}{n^{n} e^{- n} \sqrt{2 π n}} \Rightarrow

\log (\frac{3^{n}}{n!}) \sim nlog 3 - nlogn + n - \frac{1}{2} \log (2 n π)

= n (\log 3 - logn + 1 - \frac{1}{2} \log (2 n π)) \to - \infty \Rightarrow \frac{3^{n}}{n!} \to 0

also \frac{2^{n}}{n!} \to 0 \Rightarrow \lim_{n \to \infty} \frac{n! - 3^{n}}{n! - 2^{n}} = 1