In order to find a mapping from $t$ to $s$ in Laplace transform, we do the following:
$$\mathcal{L}(F(t)) = \int_0^\infty e^{-st}F(t)\,dt$$
In order to explain this question better - say we want to find a laplace transorm of $F(t) = t$. We use integration by parts:
$$\mathcal{L}(F(t)) = \int_0^\infty e^{-st}t \, dt = \left.-\frac{1}{s} e^{-st} t \right|_0^\infty - \frac{1}{s}\int_0^\infty e^{-st} \, dt$$
Now, in my book (Phil Dyke, intro to Laplace transforms and Fourier series) it is assumed that: $e^{-st}t\Big|_0^\infty = 0$. Therefore, it says that $\lim_{t \rightarrow \infty}e^{-st} = 0$.
Why is that? There is no where mentioned in my book that $s > 0$ in Laplace transform. I cannot understand why the above is assumed, and what is even $s$? If it is just another variable, is it complex/real, does it have to be only positive? If it is complex, what does it even mean if the limit in $t \rightarrow \infty$? Or am I missing something here?