How to find this limit: $\lim_{x \rightarrow \infty}{(\frac{3^x}{x!}+1)}$

Question

Can you help me please how to find this limit:

$$\lim_{x \rightarrow \infty}{(\frac{3^x}{x!}+1)}$$

Do you know the (asymptotic) behaviour of $\tfrac{a^x}{x!}$? — StackTD, Apr 05 '19 at 12:17

Minus One-Twelfth · Accepted Answer · 2019-04-05T12:17:59.530

1

Hint: Recall that (for any constant $c$), we have $$\lim\limits_{n\to\infty}\frac{c^n}{n!}=0.$$

edited Apr 05 '19 at 12:17

answered Apr 05 '19 at 12:17

Minus One-Twelfth

score 0 · Answer 2 · answered Apr 05 '19 at 12:25

0

Hint: Show that $3^n<(n-1)!$ for $n\ge9$

answered Apr 05 '19 at 12:25

egreg

2 Answers2