TITLE: Evaluating: $\lim_{n\to\infty} \frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\frac{\pi}{2}} (\sin x+\cos x)^n dx $
QUESTION [8 upvotes]: I'm supposed to compute the following limit:
$$\lim_{n\to\infty} \frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\frac{\pi}{2}} (\sin x+\cos x)^n dx  $$
I'm looking for a resonable approach in this case, if possible. Thanks.

REPLY [7 votes]: Laplace's method yields
$$\frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\pi/2} (\sin x+\cos x)^n \mathrm dx=\int_{-\pi\sqrt{n}/4}^{\pi\sqrt{n}/4}\left(\cos(t/\sqrt{n})\right)^n\mathrm dt\to\int_{-\infty}^{+\infty}\mathrm e^{-t^2/2}\mathrm dt=\sqrt{2\pi}.
$$