The points $\frac{1}{k\pi}$, where $k \in \mathbb{Z}$ are all singularities of the function $f(z) = \cot\left(\frac{1}{z} \right) - \frac{1}{z}$.
My textbook seems to think that they are simple poles of $f(z)$, but for any $z = \frac{1}{k \pi}$, the limit d.n.e. (we get $\infty$ approaching along the positive real axis, and $-\infty$ approaching along the negative real axis). According to a theorem I have on the classification of singular points, if $z_{0}$ is an isolated singular point of $f(z)$ and $\lim_{z \to z_{0}}f(z)$ does not exists, then $z_{0}$ is an essential singularity.
So, I would argue that the $\frac{1}{k\pi}$ are essential singular points, and not poles.
Who is right? Me or the book? And why?
The book is right. The correct statement is that $z_0$ is a pole of $f$ if and only if $\lim\limits_{z\to z_0} f(z) = \infty$ in the extended complex plane, which is equivalent to $\lim\limits_{z\to z_0} |f(z)| = \infty$.