This is a solution to $f(t) = \Theta(t) - 1$ found in my textbook.
$$ Lf(s) = \int^{+\infty}_{-\infty} e^{-st}(\Theta(t) - 1) dt = \int^0_{-\infty} e^{-st} (-1) dt = \left[ \frac{e^{-st}}{s} \right]^0_{-\infty} = \frac{1}{s} \left(1 - lim_{a \rightarrow -\infty} e^{-as} \right) = \frac{1}{s} $$
The thing that bothers me is that $\lim_{a -> -\infty} e^{-as} = 0$ in the solution but for me (and wolframalpha) it should be $\infty$.
I've encountered a similair problem at on other exercise, which means I think I've wrong somewhere. Please explain why the lim-term is 0.
The laplace transform is $\frac{1}{s}$, but you forgot the region of convergence for the integral, says $\Re(s) < 0 .$ Actually,
$$\lim \limits_{t \rightarrow -\infty}e^{-st} = 0, \hspace{3mm}\text{for}\hspace{3mm}\Re(s) < 0. $$