Problem finding limit - which function is asymptotically larger

Question

I have a homework question, so please don't answer fully but I would appreciate a push in the right direction.

Basically we need to figure out if $n^{n+\frac{1}{2}}e^{-n}$ is larger,smaller, or equal to $n!$ as $n$ goes to infinity.

I checked the answer, and it turns out they are equal. when $n$ goes to infinity, one divided by the other does not yield $0$. But how do I show it?

Answer 1

My suggestion is to take logs of both sides, and for $n!$ consider the integral of $\log x.$