Definition:Limit of Function (Normed Vector Space)

Definition

Let $M_1 = \struct {X_1, \norm {\,\cdot\,}_{X_1}}$ and $M_2 = \struct {X_2, \norm {\,\cdot\,}_{X_2}}$ be normed vector spaces.

Let $c$ be a limit point of $M_1$.

Let $f: X_1 \to X_2$ be a mapping from $X_1$ to $X_2$ defined everywhere on $X_1$ 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

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \norm {x - c}_{X_1} < \delta \implies \norm {\map f x - L}_{X_2} < \epsilon$

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

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \map f {\map {B_\delta} {c; \norm {\, \cdot \,}_{X_1} } \setminus \set c} \subseteq \map {B_\epsilon} {L; \norm {\, \cdot \,}_{X_2} }$.

where:

$\map {B_\delta} {c; \norm {\,\cdot\,}_{X_1} } \setminus \set c$ is the deleted $\delta $-neighborhood of $c$ in $M_1$

$\map {B_\epsilon} {L; \norm {\, \cdot\,}_{X_2} }$ 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.

This is voiced:

the limit of $\map f x$ as $x$ tends to $c$.

Equivalence of Definitions

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

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 $c$