Possible Duplicate:
Frullani proof integrals
Let $f:\left[ {0,\infty } \right] \to \mathbb R$ be a a continuous function such that $$ \mathop {\lim }\limits_{x \to0+ } f\left( x \right) = L $$Prove that $$ \int\limits_0^{\infty} {\frac{{f\left( {ax} \right) - f\left( {bx} \right)}} {x}}dx $$ converges and calculate the value.
It is known that $\int_a^\infty (f(x)/x)\,\mathrm{d}x$ converges for all a>0, but nothing of $\lim\limits_{x\to\infty}f(x)$ is told.
Also, what if $a>b$ or $a?