Definition:Limit of Mapping between Metric Spaces

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 the following equivalent conditions hold:

$\epsilon$-Ball Condition
This is voiced:
 * the limit of $f \left({x}\right)$ as $x$ tends to $c$.

Equivalence of Definitions
These definitions are seen to be equivalent in Equivalence of Definitions of Limit of Function in Metric Space.

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 $c$