# Evaluate the following limit: $\lim\limits_{ n\to\infty}\frac{(2n)!\sqrt n}{2^{2n}\cdot (n!)^{2}}$

Evaluate $$\lim_{n \to +\infty} \frac{(2n)!\sqrt n}{2^{2n}\cdot (n!)^{2}}$$.

Hint: using the Stirling approximation:$$n!=\left(\frac n e\right)^n\sqrt{2\pi n}$$ one easily finds that the limit is$$\frac1{\sqrt\pi}.$$