Definition:Limit of Mapping between Metric Spaces/Epsilon-Ball Condition

Definition
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ 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 $L \in M_2$.

$f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $c$ and is written:
 * $f \left({x}\right) \to L$ as $x \to c$

or
 * $\displaystyle \lim_{x \to c} f \left({x}\right) = L$

iff:
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \left({B_\delta \left({c; d_1}\right) \setminus \left\{{c}\right\}}\right) \subseteq B_\epsilon \left({L; d_2}\right)$.

where:
 * $B_\delta \left({c; d_1}\right) \setminus \left\{{c}\right\}$ is the deleted $\delta $-neighborhood of $c$ in $M_1$
 * $B_\epsilon \left({L; d_2}\right)$ 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 $f \left({x}\right)$ as $x$ tends to $c$.

Also known as

 * $f \left({x}\right)$ tends to the limit $L$ as $x$ tends to $c$

can also be voiced as:
 * $f \left({x}\right)$ approaches the limit $L$ as $x$ approaches $c$

Also see

 * Equivalence of Definitions of Limit of Function in Metric Space