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 real number line $\R$ under the usual (Euclidean) metric forms a metric space
- The complex plane $\C$ under the usual metric forms a metric space
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.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): limit (of $\map f x$)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): tend to