Existence of the limit of a bounded analytic function

Question

Let $X$ be a Banach space and let $U\subset\mathbb{C} $ be open. If we have an analytic function $f:U\setminus \left\{ 0\right\} \rightarrow X$ such that $\underset{x\in U\setminus \left\{ 0\right\} }{\sup }\left\Vert f\left( x\right) \right\Vert <\infty ,$ can we say that the limit $\underset{x\rightarrow 0}{\lim }f\left( x\right) $ exists. Thank you !