Question 133
Consider a continuous function $f : [0, 1] \to \mathbb{R}$ with $f(1) = 0$. Prove that the following limit exists and calculate its value:
$$
\lim_{t \to 1^-} \left( \frac{1}{1-t} \cdot \int_{0}^{1} x(f(tx) - f(x)) \, dx \right).
$$
$$
\lim_{t \to 1^-} \left( \frac{1}{1-t} \cdot \int_{0}^{1} x(f(tx) - f(x)) \, dx \right).
$$