Near an isolated singularity: bounded means removable, modulus tending to infinity means pole, and neither means essential.Suggest a slogan
Let \(f\) be holomorphic on a punctured disc \(\theset{z : 0<\abs{z-p}<r}\), so that \(p\) is an isolated singularity of \(f\).
\(p\) is a removable singularity if and only if \(\abs{f}\) is bounded on some punctured disc about \(p\).
\(p\) is a pole if and only if \(\lim_{z\to p} \abs{f(z)} = \infty\).
\(p\) is an essential singularity if and only if neither of these holds.
How are removable singularities, poles and essential singularities told apart by the behaviour of $\abs{f}$?