I want to compute
$$\lim\limits_{n \to \infty}n\int_{0}^{1}(\cos x-\sin x)^ndx$$
Someone helped me find the limit of the integral, which is $0$, but now I can't figure out this one. Also tried squeeze theorem but I only get only one side of it to converge to $1$. Moreover, I'm bound to using elementary calculus.
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