Definition:Limit of Real Function/Limit at Infinity

Definition

Let $f: \R \to \R$ be a real function.

Let $L \in \R$.

Limit at (Positive) Infinity

$L$ is the limit of $f$ at infinity if and only if:

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

This is denoted as:

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

Limit at Negative Infinity

$L$ is the limit of $f$ at minus infinity if and only if:

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

This is denoted as:

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