Limit of integral of sequence of functions

Question

Let $f:\mathbb{R}_+ \to \mathbb{R}_+$ be an integrable function.

Find : $$\lim_{n \to \infty} n \int_0^1\frac{f(nx)}{1+x} \, dx$$

I've thought about using the dominated convergence theorem, but I can't seem to be able to satisfy its conditions for a random function $f$.

Answer 1

With $u=nx$ you get

$$\int_0^n\frac{f(u)}{1+\frac un} \, du$$

and notice that

$$\left\vert\frac{f(u)}{1+\frac un}\chi_{[0,n]}\right\vert\le\vert f(u)\vert$$ so you can apply dominated convergence theorem.