$f(x)$ is continuous for $x \geq 0$, and $\lim_{x \to \infty} f(x) = \ell.$
How do I show that for any $a, b > 0$, we have $$\int_0^\infty \frac{f(ax)-f(bx)} x \, dx = (f(0) - \ell)\log\frac{b}{a}$$
I was thinking something along the lines of we can express it in the form $$\lim _{m \to 0}\int_m^1 \frac{f(ax)-f(bx)}{x} \, dx + \lim _{n \to \infty}\int_1^n\frac{f(ax)-f(bx)}{x} \, dx$$ but then I'm stuck and don't know how to proceed.
