Reading the article on the Laplace Transform in Wolfram MathWorld, I found the proof that $\mathcal{L}[f'(t)] = sF(s) - f(0)$.
I understand the first and second steps, but I don't understand the third one. Why is it that $lim_{a \to \infty} [e^{-sa} f(a)] = 0$? $e^{-sa}$ does get closer to 0 when $-sa$ approaches to $\infty$, but why does $f(a)$ get closer to 0? As far as I know, $f(a)$ could be anything, so it could be possible that $lim_{x \to \infty} f(a)$ doesn't exist.
What guarantees that the limit of $f(a)$ always exists?
I hope I'm not missing some simple property of limits here.