Question

The value of \[ \lim_{n \to \infty} \int_0^1 e^{x^2} \sin(nx)\,dx \] is _________.

Hint:

When a bounded smooth function multiplies a rapidly oscillating sine or cosine term, its integral tends to zero — a direct result of the Riemann–Lebesgue Lemma.

Answer 1

Correct Answer: 0

Step 1: Note the structure.
The integrand is \(e^{x^2} \sin(nx)\), which oscillates increasingly fast as \(n \to \infty.\)
Step 2: Use the Riemann–Lebesgue Lemma.
For any continuous \(g(x)\) on \([0,1]\), \[ \lim_{n \to \infty} \int_0^1 g(x)\sin(nx)\,dx = 0. \] Here \(g(x) = e^{x^2}\) is continuous on \([0,1]\).
Step 3: Conclusion.
Thus, \[ \lim_{n \to \infty} \int_0^1 e^{x^2}\sin(nx)\,dx = 0. \]