How can we decide that the $$\exp-(s-ia)t$$ is converging or diverging at $t\to \boldsymbol{+\infty} $.
During the derivation of Laplace transform of $$\sin(at)$$ while solving limit that we get after integration we wrote $$\exp(-(s-ia)(\infty))=0,\qquad\exp((s+ia)(\infty))=0,$$ but the power is in complex form. How can we say that it is converging or diverging? If we write $$\exp(-st )\exp(iat)$$ (separately) and then put infinity then how will we decide the complex part is converging or diverging at infinity. We can not say ${a}$ is either positive or negative because two of the terms while deriving the Laplace transform of sin(x) is $$\exp(-st) \exp(ait) $$ and one is $$\exp(-st) \exp(-ait)$$ Also the term is no more a single dimensional thing, it is now dependent on two dimensions, one real and one complex.
Note that $|e^{-x+iy}| = e^{-x}$, where $x,y$ are real.
To prove $\lim_{x\to\infty} exp(-x+iy) = 0$, we need to show that for every $\epsilon > 0$, $\exists n>0$
s.t. $|e^{-x+iy} -0|< \epsilon, \forall x>n\iff e^{-x}< \epsilon, \forall x>n$, which is obvious.
So we must have $s>0$ for the limit to be $0$.