Proof that a limit can be brought into the integral

Question

I need to prove that $$ \lim\limits_{\large s \to \infty} \int \limits_{0_+}^\infty \left[ \dfrac{d x(t)}{dt} \right] e^{-\large st} dt = 0 $$

The problem is that I need to prove why I can bring the limit inside the integral for s a complex variable.

If someone can help me it would be very nice.

Answer 1

What do you know about the function $x$?

It may be helpful to note that this is the Laplace transform of $x'$, i.e.

$$ \int_0^\infty \frac{dx(t)}{dt} e^{-st} dt = \mathcal{L}\left\{x' \right\}(s) = s\mathcal{L}\left\{x\right\}(s) - x(0). $$

So, if you know some properties of the Laplace transform of $x$, then you can work with the limit directly. Otherwise, this demonstrates that this cannot hold for general $x$; e.g. if $x$ is the unit step function, then the limit is $1$.