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

Question

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

Please help with steps, Dont know how to break it down to cancel out terms.

Answer 1

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