Let $f\in C^0(\mathbb R)$,
What is : $$\lim_{x\to 0}\int_x^{2x}\dfrac{f(t)}{t}\,dt$$
I found that : $\forall x\in\mathbb R, \int_x^{2x}f(t)/2x\,dt\le\int_x^{2x}f(t)/t\,dt \le \int_x^{2x}f(t)/x\,dt$.
(It only works if $f$ is non negative.)
We can show that $\int_x^{2x}f(t)/x\,dt = \int_1^{2}f(ux)\,du \to f(0)$ when $x\to0$.
But it doesn't even end into finding if such limit exist.
Have you got some ideas ?
