Laplace transforms: Finding f(0) given F(s)

Question

So in our differential equations class we've used this property a couple times:

$f(0) = \lim_{s \to \infty} sF(s)$

where $F(s)$ is the Laplace transform of $f(t)$.

But I can't find an explanation or proof of why or when it works. If anyone could explain or link to an explanation it would be much appreciated!

Answer 1

We know that

$$\mathcal{L}\{f'(t)\} = \int_0^\infty f'(t)e^{-st}dt = sF(s) - f(0)$$

Then if one takes limits on both sides we get that

$$0 = \lim_{s\to \infty} sF(s) - f(0)$$

assuming that $f'(t)$ was bounded so we could apply the dominated convergence theorem and move the limit inside of the integral.