When evaluating the Laplace Transform of $f(t)=c$, assuming $s$ is some complex number $s=\sigma+i\omega$
$F(s)=\frac cs(1-\lim_{T\to \infty}e^{-\sigma T}(\cos\omega T-i\sin\omega T)$
I can see that $e^{-\sigma T}$ will approach zero as long as $\sigma>0$ but $\cos\omega T-i\sin\omega T$ doesn't approach anything.
I know that the limit of $f(x)\cdot g(x)$ is the limit of $f(x)$ multiplied by the limit of $g(x)$, provided that both functions have a limit.
But in this case, only one of them has a limit (equal to zero), while the other is undefined.
My textbook says that $\lim_{T\to \infty}e^{-\sigma T}(\cos\omega T-i\sin\omega T)=0$ provided that $\sigma >0$
I don't have experience evaluating limits like this, what am I missing?
Since $|e^{xi}| = 1$ for all $x \in \mathbb R$ and $|e^a| = e^a$ for all $a \in \mathbb R$, $$|e^{-\sigma T}e^{-\omega T i}| = |e^{-\sigma T}||e^{-\omega T i}| = e^{-\sigma T}$$ and this shows that the distance from $e^{-\sigma T}e^{-\omega T i}$ to the origin can be make arbitrarily small taking $T$ sufficiently large.