A holomorhic function $f $ in the unit disc such that $\lim_{z\rightarrow 1}f(z)$ does not exist

Question

Let $f$ be a holomorphic function in the open unit disc such that $\lim_{z\rightarrow 1}f(z)$ does not exist. Let $\sum_{n=0}^{\infty}a_nz^n$ be the Taylor sereis expansion of $f$ about $z=0$ and $R$ be its radius of convergence.Then

$(1)R=0$

$(2)0\lt R \lt 1$

$(3)R=1$

$(4)R\gt 1$

My thoughts or rather questions..

Can I say that $z=1$ is an essential singularity taking in concern that the limit there does not exist?

Again due to non-existence of limit at $1$, it's clear $R\le 1$ but how do I exactly determine the radius of convergence of the Taylor series about $0$?

Please help me out .A good hint can be enough for me. Thanks a lot!!