I want to figure out the process for showing why the function $\cos(1-\frac{1}{z})$ has an essential singularity at $z=0$ without using knowledge of the Laurent expansion. I know the process should be to rule out the possibility of removable singularities or poles, but do not know how to do this for this function.
Attempt I was thinking I would show since $$\lim_{z\to 0} |\cos(1-\frac{1}{z})| \text{ DNE } $$ since the function oscillates between $1$ and $-1$ for $z$ near zero for positive values, this rules out the possibility of a pole since the limit is not $\infty$ and the singularity is not removable since the limit is not finite.Is this the correct approach? What are some other ways, to show that zero is an essential singularity?