\begin{definition}[Definition:Limit of Mapping between Metric Spaces/Epsilon-Ball Condition]
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $c$ be a limit point of $M_1$.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$.
Let $L \in M_2$.
$\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $c$''' and is written:
:$\map f x \to L$ as $x \to c$
or
:$\ds \lim_{x \mathop \to c} \map f x = L$
{{iff}}:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {c; d_1} \setminus \set c} \subseteq \map {B_\epsilon} {L; d_2}$
where:
:$\map {B_\delta} {c; d_1} \setminus \set c$ is the deleted $\delta $-neighborhood of $c$ in $M_1$
:$\map {B_\epsilon} {L; d_2}$ is the open $\epsilon$-ball of $L$ in $M_2$.
That is, for every open $\epsilon$-ball of $L$ in $M_2$, there exists a deleted $\delta $-neighborhood of $c$ in $M_1$ whose image is a subset of that open $\epsilon$-ball.
This is voiced:
: '''the limit of $\map f x$ as $x$ tends to $c$'''.
\end{definition}