Finding the Limit of F(s) as s Goes to Infinity: Exploring Exponential Order

[shapiro478]

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?

 

[gel]

You can use the http://en.wikipedia.org/wiki/Dominated_convergence_theorem" . It's stated in that link for general measure spaces, but you can just replace $d\mu$ with dx for integrating over the reals.

or,

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.