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Let $f : [−1, 1] → \mathbb{R}$ be continuous. Assume that $ ∫_{-1}^1f (t)\,\mathrm dt = 1$.
Evaluate: $ \lim_{n\to\infty} ∫_{-1}^1f (t)\cos^2(nt)\,\mathrm dt$.

Totally stuck. How can I solve it? Thanks for your help!

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Write $\cos^2{n t} = \frac{1}{2} (1+\cos{2nt})$ and rewrite the integral to get

$$ \int_{-1}^{1} dt \: f(t) \cos^2{n t} = \frac{1}{2} \left ( 1+ \Re{\int_{-1}^{1} dt \: f(t)e^{i 2n t}} \right ) $$

Use integration by parts to show that the integral on the right hand side vanishes as $n \rightarrow \infty$.

EDIT

Even if integration by parts does not work (e.g., the case where $f(t) = (1-t^2)^{-\frac{1}{2}}$, this integral vanishes by the Riemann-Lebesgue Lemma.

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