Limit Problem

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Say a function f and its derivative are everywhere continuous and of exponential order at infinity. F(s) is the Laplace transform of f(x). I need to find the limit of F as s goes to infinity.

I use the integral definition of the Laplace transform and the fact that f is of exponential order. My problem is that I don't know if you can move the limit inside the integral. If you can, then it is clear that the result is 0. How can I justify this step, or is there a better approach?

Answers and Replies

You can use the" [Broken]. It's stated in that link for general measure spaces, but you can just replace [itex]d\mu[/itex] with dx for integrating over the reals.


if f is bounded by exp(ax), then bound f(x)exp(-sx) by exp(-(s-a)x). Use this to bound F(s) and you can calculate the rate at which it goes to 0.
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