Definition of a Complex Limit

Let $z_0\in \mathbb{C},f:D\to \mathbb{C}$ where $D\subset C$ containing some neighbourhood $N_{\delta }(z_0)$ of $z_0$ without necessarily including $z_0$.

We say that $f$ as the limit $l$ as $z\to z_0$ and denote ${\lim }_{z\to z_0}f(z)=l$ when, for every $ϵ>0$ there is a $\delta >0$, such that for all $z\in N_{\delta }(z_0):z\ne z_0$ we have $|f(z)-l|<ϵ$.