sequences and series - Using the squeeze theorem liberally to prove a limit

The question is, find the limit of $\ a_n = (e^n+3^n)/5^n$ as $\ n→∞ $
I tried using L'hopital's rule, but didn't seem to get anything useful, so I figured I may be able to use the squeeze theorem. Would this be an appropriate use of the squeeze theorem, and would there be a better method to proving that the limit approaches zero?

$\ 1/n ≤ (e^n+3^n)/5^n ≤ 10/n $ for all n, with both $\ 1/n $ and $\ 10/n $ approaching zero as n approaches infinity.

Answer

Hint: I don't believe the bounds you have are correct. Instead, I suggest something like $$0\leq\frac{e^n+3^n}{5^n}\leq\frac{3^n+3^n}{5^n}=2\cdot\frac{3^n}{5^n}.$$ These inequalities are much more obvious, hold for all $n$, thus we can take the limit. What do we get?