\begin{definition}[Definition:Limit of Complex Function]
The definition for the limit of a complex function is exactly the same as that for the general metric space.
Let $A_1, A_2 \subseteq \C$ be subsets of the complex plane.
Let $c$ be a limit point of $A_1$.
Let $f: A_1 \to A_2$ be a complex function from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$.
Let $L \in A_2$.
Then $\map f z$ is said to '''tend to the limit $L$ as $z$ tends to $c$''', and we write:
:$\map f z \to L$ as $z \to c$
or
:$\ds \lim_{z \mathop \to c} \map f z = L$
if the following equivalent conditions hold.
This is voiced:
: '''the limit of $\map f z$ as $z$ tends to $c$'''.
\end{definition}