Reading on control theory and the Laplace transform of the unit step function, I came upon the following in my textbook.
The Laplace transform defined as: $$Y(s)=\int_{0}^{\infty}y(t)e^{-st}dt$$
s being a complex number $s=\mu+j\omega$
Given the unit step function $\sigma(t)=\begin{cases} 1 & t\ge0\\ 0 & t<0 \end{cases} $
The Laplace transform of $\sigma$ is
$$ Y(s)=\int_{0}^{\infty}1e^{-st}dt $$
$$ =-\left.\frac{e^{-st}}{s}\right|_{0}^{\infty}=\frac{1}{s} $$
My question is about the last part and how to calculate it step by step. I tried and did the following:
$$ -\left.\frac{e^{-st}}{s}\right|_{0}^{\infty}=\lim_{t\rightarrow\infty}-\frac{e^{-s\cdot t}}{s}-(-\frac{e^{-s\cdot0}}{s}) $$
$$ =-\lim_{t\rightarrow\infty}\frac{1}{s\cdot e^{s\cdot t}}+\frac{e^{0}}{s} $$
$$ =-\lim_{t\rightarrow\infty}\frac{1}{s\cdot e^{s\cdot t}}+\frac{1}{s} $$
Since the right term is $\frac{1}{s}$, which is the end result, I conclude the left term must become 0.
If $s$ was a real number I would have no trouble with this, but I wasn't certain how to do this when $s$ is a complex number. Trying with Wolfram Alpha, it says this expression has value zero , but doesn't specify if using real or complex numbers.