Definition:Limit of Function (Normed Vector Space)/Epsilon-Delta 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
 * $\displaystyle \lim_{x \to c} \map f x = L$

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

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 $f$.

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