Definition:Limit of Function (Normed Vector Space)/Epsilon-Ball Condition

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


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

where:
 * $\map {B_\delta} {c; \norm {\,\cdot\,}_{X_1}} \setminus \set c$ is the deleted $\delta $-neighborhood of $c$ in $M_1$
 * $B_\epsilon \left({L; \norm {\,\cdot\,}_{X_2}}\right)$ 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 $c$.

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 $c$

Also see

 * Equivalence of Definitions of Limit of Function in Normed Vector Space