Definition:Limit of Real Function/Limit at Infinity

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

Let $L \in \R$.

$L$ is the limit of $f$ at infinity :
 * $\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x > c : \left\vert{f \left({x}\right) - L}\right\vert < \epsilon$

This is denoted as:
 * $\displaystyle \lim_{x \mathop \to \infty} f \left({x}\right) = L$

$L$ is the limit of $f$ at minus infinity :
 * $\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x < c : \left\vert{f \left({x}\right) - L}\right\vert < \epsilon$

This is denoted as:
 * $\displaystyle \lim_{x \mathop \to - \infty} f \left({x}\right) = L$