Definition:Limit of Mapping between Metric Spaces/Epsilon-Delta Condition
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Definition
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
- $\ds \lim_{x \mathop \to c} \map f x = L$
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \map {d_1} {x, c} < \delta \implies \map {d_2} {\map f x, L} < \epsilon$
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 $f$.
This is voiced:
- the limit of $\map f x$ as $x$ tends to $c$.
Also known as
- $\map f x$ tends to the limit $L$ as $x$ tends to $c$
can also be voiced as:
- $\map f x$ approaches the limit $L$ as $x$ approaches $c$