Limit of a integral For $f:[0,1]\rightarrow \mathbb{R}$ continuous, let $a >0.$
\begin{eqnarray*}
L=\lim_{\varepsilon \rightarrow 0}\int_{\varepsilon a}^{\varepsilon b} \frac{f(x)}{x} dx
\end{eqnarray*}
Show that $L=f(0)\ln(\frac{b}{a})$.The first I thought was in mean value theorem for integrals. but  I don't know how can start, if can give me some hint to start. I will be gratefull.

By the substitution $y=\frac x {\epsilon}$ the integral becomes $\int_a^{b} \frac {f(\epsilon x)} x dx$. Can you finish?
