Problem

Source: Romanian National Olympiad 2000, Grade XI, Problem 1

Tags: real analysis, integrals, limits, continuity, calculus, integration



Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ a) Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. b) Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $