How does one PROVE that lim as n->∞[(5^n)/n!] = 0

I was asked to find this limit, and knew it was 0, since n! will obviously surpass 5^n if n->∞. But how can I PROVE this?

Here is a Squeeze Theorem proof.

Note that for n > 5, we have
5^n/n! = 5^n/(1 * 2 * 3 * 4 * 5 * 6 * 7 * ... * n)
.........< 5^n/(1 * 2 * 3 * 4 * 5 * 6 * 6 * ... * 6)
.........= 5^n/(1 * 2 * 3 * 4 * 5 * 6^(n-5)
.........= (6^5 / 5!) (5/6)^n.

So, we have 0 < 5^n/n! < (6^5 / 5!) (5/6)^n for all n > 5.
Since lim(n→∞) 0 = 0 = lim(n→∞) (6^5 / 5!) (5/6)^n, we conclude that lim(n→∞) 5^n/n! = 0.

I hope this helps!

Google "ratio test limit" and you'll find all you need to know and more.