TITLE: How to find $\lim\limits_{n \to \infty}\int_{0}^{1}(\cos x-\sin x)^ndx$?
QUESTION [7 upvotes]: I want to compute $$\lim\limits_{n \to \infty}\int_{0}^{1}(\cos x-\sin x)^n dx.$$
I tried the squeeze theorem, tried simplifying the integral, but I eventually got nothing.

REPLY [2 votes]: Because $(\cos(x)-\sin(x)) \leq 1-x$ over the range of integration, an upper bound for the integral is
$$\int_{0}^{1} (1-x)^n dx.$$
But,
$$\lim_{n \rightarrow \infty} \int_{0}^{1} (1-x)^n dx = \lim_{n \rightarrow \infty} \frac{1}{n+1} = 0.$$
Because the original integral is positive for all $n$, the limit must be zero by the Squeeze theorem.