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Bringing a limit inside of an integral

  1. Mar 13, 2012 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    3. The attempt at a solution

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