Show that if z0 is an isolated singularity of f(z) that is not removable, then z0 is an essential singularity for ef(z).
z0 is a pole of f(z) of order N iff f(z) = g(z)/(z-z0)^N, where g is analytic at z0 and g(z0) is not 0, iff 1/f(z) is analytic at z0 and has a zero of order N, iff |f(z)| → ∞ as z → z0.
The Attempt at a Solution
I know what to do if z is an essential singularity. But I'm having trouble if z is a pole. It seems that what I must do is find a sequence of points going to z0 such that f(z) goes to infinity, and also a sequence of points going to z0 such that f(z) approaches some complex number. I don't know how to do that.