$f:[-1,1]\to\Bbb{R}$ be continuous and $\int_{-1}^{1}f(t)dt=1$ then find $\displaystyle \lim_{n\to \infty} \int f(t) \cos^2nt \mathrm{d}t$.[NBHM2008]

Question

let $f:[-1,1]\to \mathbb{R}$ be a continuous function such that $\int_{-1}^{1}f(t)dt=1$ then find $\displaystyle \lim_{n\to \infty} \int f(t) \cos^2nt \mathrm{d}t$.
My method: I have solved the problem by choosing $f(t) =\frac12$. And the answer I get is also $\frac12$. But I want to know, without choosing this type of example, is there any other general proof? Thanks in advance

Answer 1

$\int f(t)\cos^{2}(nt)dt=\frac 1 2 \int f(t)[1+\cos (2nt)]dt$. By Riemann Lebesgue Lemma $\int f(t) \cos (2nt)dt \to 0$, so the answer is $\frac 1 2$.