Definition:Limit of Mapping between Metric Spaces/Epsilon-Delta 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:
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < d_1 \left({x, c}\right) < \delta \implies d_2 \left({f \left({x}\right), L}\right) < \epsilon$

That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that every point in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some point $L$ in the codomain of $f$.

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