Problem

Source: Romanian National Olympiad 2024 - Grade 12 - Problem 3

Tags: function, calculus, integration



Let $f:[0,1] \to \mathbb{R}$ be a continuous function with $f(1)=0.$ Prove that the limit $$\lim_{t \nearrow 1} \left( \frac{1}{1-t} \int\limits_0^1x(f(tx)-f(x)) \mathrm{d}x\right)$$exists and find its value.