What Happens When Evaluating Improper Integrals with Limit?

[JIHUN]

Homework Statement

Evaluating the following formula:

The Attempt at a Solution

Since the integral part is unknown, dividing the case into two: converging and diverging
If converging: the overall value will always be 0
If diverging: ...?

 

[Sir Beaver]

As you say, if the integral converges for $x \to 0^+$, the result is always zero. However, even if the integral diverges, it is still possible that one can get a finite value. This depends on the speed of divergence. Try it out with $f(t) = 1$ and $f(t) = 1/t$. What happens for $f(t) = 1/t^\alpha, \alpha > 1$?