MathDB
int f <1 implies y=0 is asymptote of id.f, where f is nonincreasing

Source: Romanian District Olympiad 2009, Grade XII, Problem 1

October 8, 2018
functioninequalitiesIntegralreal analysis

Problem Statement

Let f:[0,∞)⟶[0,∞) f:[0,\infty )\longrightarrow [0,\infty ) a nonincreasing function that satisfies the inequality: \int_0^x f(t)dt <1, \forall x\ge 0. Prove the following affirmations:
a) ∃lim⁡x→∞∫0xf(t)dt∈R. \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} .
b) lim⁡x→∞xf(x)=0. \lim_{x\to\infty} xf(x) =0.
Back to ProblemsView on AoPS