I would like to know whether the following statement is true or not
If $f$ is an analytic function satisfied $|f(z)|\to\infty$ when $z\to z_0$, then $f$ is a pole at $z=z_0$
I would say the statement is false.
By a theorem in my textbook, for $z=z_0$ be a pole of $f$, it requires $$lim_{z\to z_0}f(z)=\infty$$
We know that $lim_{z\to z_0}f(z)=L$ implies $lim_{z\to z_0}|f(z)|=|L|$ (the proof is skipped), but the reverse may not be true. So $lim_{z\to z_0}|f(z)|=\infty$ may not give $lim_{z\to z_0}f(z)=\infty$. That is, we cannot conclude that it is a pole.
Can I say in this way?