Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function such that $\lim _{z \rightarrow 0}\left|f\left(\frac{1}{z}\right)\right|=\infty$. Then which of the

following is true?

1. $f$ is constant

2. $f$ can have infinitely many zeros

3. $f$ can have at most finitely many zeros

4. $f$ is necessarily nowhere vanishing