RO-NMO 2014 Problem 123

Let $f:[1,\infty )\longrightarrow (0,\infty )$ be a continuous function satisfying the following properties: $\text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}}$ $\text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}.$ a) Show that $\lim_{x\to\infty } \frac{f(x)}{x}=0.$ b) Prove that $\lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0.$