# Bringing a limit inside of an integral

1. Mar 13, 2012

### Random Variable

1. The problem statement, all variables and given/known data

Under what conditions does $\lim_{s \to 0^{+}} \int_{0}^{\infty} f(x) e^{-sx} \ dx = \int_{0}^{\infty} f(x) \ dx$ ?

2. Relevant equations

3. The attempt at a solution

If justification is ever offered, it's that $\int_{0}^{\infty} f(x) \ dx$ converges. But I'm not certain that's enough justification. It was suggested to me to also show that $\int_{0}^{\infty} f(x) e^{-sx} \ dx$ converges absolutely.