# Help with Limit: $\lim_{n \to \infty} \frac{(2 n)! e^n (n)^n}{n! (2 n)^{2 n}}$

Help with Limit: $$\lim_{n \to \infty} \frac{(2 n)! e^n (n)^n}{n! (2 n)^{2 n}}$$

EDIT: Posted a similar question just yesterday but bad formating from my part dropped a term, and lead me to strange places. Honestly, I am just plain stuck, I have been hitting my head against it for $2$ days straight. I know the solution should be $2^{-1/2}$ but...

Help would be appreciated. If anyone has a Wolfram Alpha PRO account I am sure that thing churns out the solution as it is something quite basic I just don't see it and its driving me crazy.

• Wanted to delete this one, but the community was way too fast. – Katpton Liamfuppinshire Nov 8 '16 at 6:59

Hint: Use Stirling's approximation $$n!\approx n^n e^{-n}\sqrt{2\pi n}$$ for large $n$.
Hint. Use Stirling approximation: $$n!\sim\sqrt{2\pi n} \cdot\frac{n^n}{e^n}.$$ which implies that $$(2n)!\sim2\sqrt{\pi n} \cdot\frac{(2n)^{2n}}{e^{2n}}.$$