When taking Laplace transforms such as
$$\int_{t_0}^\infty e^{-st},$$
we subsequently get
$$-\dfrac{1}{s} \left[ e^{-st} \right]^\infty_{t_0}.$$
Now, my textbook author will just claim that $e^{-s \infty} = 0$, leaving us with the solution $\dfrac{1}{s} e^{-s t_0}$. However, since $s$ is a complex number, this seems exceedingly handy-wavy to me. I find this unacceptable, and I want to understand the mathematics of what's going on here.
In my research, I have encountered the concept of radius of convergence, and I suspect that this has something to do with what's going on here. However, doing a search for keywords, my textbook does not address the radius of convergence in any direct context of the Laplace transform, but rather later in the context of the Taylor/Laurent series.
I would greatly appreciate it if people could please take the time to explain what's going on here. Please note that I have not studied complex analysis - just pieces of it - so please provide careful explanations.