Determine the character of the singularity at $z=0$ for each of the following functions: \begin{align} &\frac{1/z^{7}}{e^{z}-1} \tag{1} \\ &\frac{}{} \\ &\frac{e^{\frac{1}{z}}-1}{z^{7}} \tag{2} \\ &\frac{}{} \\ &\frac{e^{z^{4}}-1}{z} \tag{3} \end{align}
I am currently reading Fundamentals of Complex Analysis with Applications to Engineering and Science by Saff & Snider. This book provides a "summary" of the various equivalent characterizations of the three types of isolated singularities on p.284, theorem 18.
Based on this, I argue $(2)$ is an essential singularity since $|f(z)|$ neither is bounded near $0$ nor goes to infinity as $z \to 0$. Moreover, I claim $(1)$ will produce a pole, for $|f(z)| \to \infty$ as $z \to 0$. Finally, I claim $(3)$ is a removable singularity for $f(z)$ has a limit as $z \to 0$, namely $0$.