How to find $\lim\limits_{n \to \infty}\int_{0}^{1}(\cos x-\sin x)^ndx$?

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.