# Definition:Limit of Mapping

## Limit of Mapping between Metric Spaces

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:

$\ds \lim_{x \mathop \to c} \map f x = L$

if and only if the following equivalent conditions hold:

### $\epsilon$-$\delta$ Condition

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \map {d_1} {x, c} < \delta \implies \map {d_2} {\map f x, L} < \epsilon$

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 $f$.

### $\epsilon$-Ball Condition

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {c; d_1} \setminus \set c} \subseteq \map {B_\epsilon} {L; d_2}$

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.

## Real and Complex Numbers

As:

the definition holds for sequences in $\R$ and $\C$.

However, see the definition of the limit of a real function below:

## Limit of Real Function

The concept of the limit of a real function has been around for a lot longer than that on a general metric space.

Let $\openint a b$ be an open real interval.

Let $c \in \openint a b$.

Let $f: \openint a b \setminus \set c \to \R$ be a real function.

Let $L \in \R$.

### Definition 1

$\map f x$ tends to the limit $L$ as $x$ tends to $c$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$

where $\R_{>0}$ denotes the set of strictly positive real numbers.

### Definition 2

$\map f x$ tends to the limit $L$ as $x$ tends to $c$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: x \in \map {N_\delta} c \setminus \set c \implies \map f x \in \map {N_\epsilon} L$

where:

$\map {N_\epsilon} L$ denotes the $\epsilon$-neighborhood of $L$
$\map {N_\delta} c \setminus \set c$ denotes the deleted $\delta$-neighborhood of $c$
$\R_{>0}$ denotes the set of strictly positive real numbers.

## Complex Analysis

The definition for the limit of a complex function is exactly the same as that for the general metric space.

## Also see

• Results about limits of mappings can be found here.