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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?

  • 0
    I think that the value of the integral depends on $f$.2012-12-30
  • 0
    Is it an assumption that $\int_a^{\infty} \frac{f(x)}{x} \, dx$ converges...? Because in general that's not true (take $f(x):=x^2$).2012-12-30
  • 0
    yes, an assumption, part of the assignment.2012-12-30

2 Answers 2