Question 68
Let $f : [0, 1] \to \mathbb{R}$ be a continuous function.
a) Show that
$$
\lim_{n \to \infty} \int_{0}^{1} f(x^n) dx = f(0).
$$
b) If $f(0) = 0$ and $f$ is differentiable at the right in $0$, show that the limits exist, are finite and equal.
$$
\lim_{\varepsilon \to 0} \int_{\varepsilon}^{1} \frac{f(x)}{x} dx \quad \text{and} \quad \lim_{n \to \infty} \left( n \cdot \int_{0}^{1} f(x^n) dx \right)
$$
a) Show that
$$
\lim_{n \to \infty} \int_{0}^{1} f(x^n) dx = f(0).
$$
b) If $f(0) = 0$ and $f$ is differentiable at the right in $0$, show that the limits exist, are finite and equal.
$$
\lim_{\varepsilon \to 0} \int_{\varepsilon}^{1} \frac{f(x)}{x} dx \quad \text{and} \quad \lim_{n \to \infty} \left( n \cdot \int_{0}^{1} f(x^n) dx \right)
$$