RO-NMO 2017 Problem 124

A function $f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}$ has the property that $\lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1.$ a) Give an example of what $f$ could be if it's continuous and $f/\text{id.}$ doesn't have a limit at $\infty .$ b) Prove that if $f$ is nondecreasing then $f/\text{id.}$ has a limit at $\infty ,$ and determine it.