I want to classify the singularity at $z=0$ of the function
My book says, the answer is pole.
I know that the options are either, it could be a isolated removable singularity, a pole, or an essential singularity.
But here is what I thought; the limit as $z$ goes to $ 0$ is clearly $2$ via L'Hospital rule, but my notes also say that p is a pole if the limit as we approach the absolute value of the function is infinity. So I am having trouble seeing why this is a pole.
On the other hand, since the limit is 2, there will certainly exist a $r>0$ such that $|f(z)| \le 2$ for all $z$ in the punctured disk around it.
So that leads me to believe that we are dealing with a removable singularity..
Is that correct? Or the book correct and I am missing something cruicial?