Does an impulse function at t=0 go to infinity, or 1?
The \$\delta(t)\$ is defined as positive infinite amplitude and infinitesimal width with an area of 1 at \$\delta(0)\$, and 0 otherwise.

As Wikipedia states: 

\$\int^\infty_{-\infty}\delta(t)~dt = 1\$

Since it only has one point that has a non-zero value, which is at \$\delta(0)\$, then that is the only point we need to consider. And the area at that point is 1 per definition.



And \$x^0=1\$ which is why \$e^{-s×0}=1\$