# Definition:Limit of Real Function/Definition 1

## 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 $c$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$

where $\R_{>0}$ denotes the set of strictly positive real numbers.

## Notation

$\map f x$ tends to the limit $L$ as $x$ tends to $c$, is denoted:

$\map f x \to L$ as $x \to c$

or

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

The latter is voiced:

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