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Real analysis: limits of f/id, F/id and more

Source: Romania National Olympiad 2014, Grade XII, Problem 3

March 3, 2019
functionreal analysis

Problem Statement

Let f:[1,∞)⟶(0,∞) f:[1,\infty )\longrightarrow (0,\infty ) be a continuous function satisfying the following properties:
(i)∃lim⁡x→∞f(x)x∈R‾ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} (ii)∃lim⁡x→∞1x∫1xf(t)dt∈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→∞f(x)x=0. \lim_{x\to\infty } \frac{f(x)}{x}=0. b) Prove that lim⁡x→∞1x2∫1xf2(t)dt=0. \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0.
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