IMC 1994 · Problem 5

a) Let $f \in C[0,b]$, $g \in C(\mathbb{R})$ and let $g$ be periodic with period $b$. Prove that $\int_0^b f(x)g(nx)\,dx$ has a limit as $n \to \infty$ and

$$\lim_{n \to \infty} \int_0^b f(x)g(nx)\,dx = \frac{1}{b} \int_0^b f(x)\,dx \cdot \int_0^b g(x)\,dx.$$

b) Find

$$\lim_{n \to \infty} \int_0^{\pi} \frac{\sin x}{1 + 3\cos^2 nx}\,dx.$$