Find each of the following limits:

(1) $\lim_{z \to 0}{e^z}\ =\ $

(2) $\lim_{z \to 2\pi i}{e^z-e^{-z}}\ =\ $

(3) $\lim_{z \to \frac{\pi i}{2}}{(z\ +\ 2)e^z}\ =\ $

(4) $\lim_{z \to \pi i}{\exp(\frac{z^2\ +\ \pi^2}{z\ +\ \pi i})}\ =\ $