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?