The Second Problem
Hello guys! This is Arnab and you're gonna be facing the next problem from BMC!!Here's the problem:Let \(g:[0,1] \rightarrow \mathbb{R}\) be a continuous function in its domain such that $$|\lim_{x\to 0^{+}} \frac{g(x)}{x}|<\infty$$Show that for all continuous functions \(f:[0,1]\rightarrow \mathbb{R}\),$$\lim_{n \to \infty} n \int_0^1 f(x)g(x^n) dx = f(1) \int_0^1 \frac{g(x)}{x} dx$$ Please make sure to mention your thoughts on this problem in the comments. We will be releasing the solutions tomorrow again. So stay tuned!