Suppose $f:[a,b]\to\mathbb{R}$ is continuous. Determine if the following limit exists
$$\lim_{n\to\infty}\int_a^b f(x)\sin^3{(nx)} \:dx.$$
As $f(x)$ and $\sin^3{(nx)}$ are continuous, so their product is Riemann integrable. However $\lim_{n\to\infty} f(x)\sin^3{(nx)} $ does not exist, so it's not uniformly convergence and we cannot pass the limit inside the integral. It also doesn't satisfy in the conditions of Dini Theorem. I don't know how to make a valid argument for this problem, but I think by what I said the limit doesn't exist. I appreciate any help.
Best Answer
Riemann-Lebesgue lemma. Note that $\sin^3(nx) = \frac{3}{4} \sin(nx) - \frac{1}{4} \sin(3nx)$.