Limit of an integral
Posted: Sun Aug 06, 2017 6:43 am
The following exercise is just an alternative of IMC 2017/2/1 problem. It is quite easy but it's not a bad idea to have it here as well.
Given the continuous function $f:[0, +\infty) \rightarrow \mathbb{R}$ such that $\lim \limits_{x \rightarrow +\infty} x^2 f(x) = 1$ prove that
$$\lim_{n \rightarrow +\infty} \int_0^1 f(n x)\, {\rm d}x =0$$
Given the continuous function $f:[0, +\infty) \rightarrow \mathbb{R}$ such that $\lim \limits_{x \rightarrow +\infty} x^2 f(x) = 1$ prove that
$$\lim_{n \rightarrow +\infty} \int_0^1 f(n x)\, {\rm d}x =0$$