How to solve the limit of $\lim_{n\to\infty}\frac{n^{n + 1/2}}{n!e^{n}}$ as $n$ goes to infinity?

Question

I found out from wolfram alpha that $$\lim_{n\to\infty}\frac{n^{n + 1/2}}{n!e^{n}}=\frac{1}{\sqrt{2\pi}}$$ However, I don't know how to get to this answer.

I've tried l'hopitals, but that doesn't simplify the fraction at all. How do you calculate this limit?

Answer 1

As mentioned in the comments by @Sine of the Time, one has that:

\begin{align*} \lim_{n\to\infty}\frac{n^{n + 1/2}}{n!e^{n}} & = \lim_{n\to\infty}\frac{n^{n + 1/2}}{n^{n}e^{-n}e^{n}\sqrt{2\pi n}} = \lim_{n\to\infty}\frac{1}{\sqrt{2\pi}} = \frac{1}{\sqrt{2\pi}} \end{align*}