The value of $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$

Question

Using the fact $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos nx dx=0$ ,find the value of $$\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$$

I tried through integrating by parts , also through the $1^{st}$ Mean Value Theorem of integral calculus but I c

Answer 1

$$\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx \,dx = \lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\frac{\cos 2nx + 1}{2} \,dx$$

Answer 2

hint:

$$\cos(2t) =2\cos^2 t - 1$$