How to calculate $\lim_{n\to\infty} \frac{2^n}{n!}$?

Question

How to calculate $\lim_{n\to\infty} \frac{2^n}{n!}$? I tried using $\lim \frac{x^n}{a^x} = 0$ but it didn't work

Answer 1

The limit is $0$, since factorial growth is faster than exponential growth.

$$ 2^n/n! = (2/1)*(2/2)*\ldots*(2/n)$$ $$ \leq 2 * 2/n$$

This quantity can clearly be made arbitrarily small for sufficiently large n, hence the limit is 0 (since $2^n/n!$ is positive)

Answer 2

We have $$2^n = (1 + 1)^n = \sum_{k = 0}^n\frac{n!}{k!(n-k)!}.$$ Try to conclude from that.