Definition:Limit of Real Function/Definition 2

Definition
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 $L \in \R$.

$\map f x$ tends to the limit $L$ as $x$ tends to $c$ :
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: x \in \map {N_\delta} c \setminus \set c \implies \map f x \in \map {N_\epsilon} L$

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.

Also see

 * Equivalence of Definitions of Limit of Real Function