# Definition:Limit of Real Function/Limit at Infinity/Negative

## Definition

Let $f$ be a real function defined on an open interval $\openint \gets a$.

Let $L \in \R$.

$L$ 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:

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

### Increasing Without Bound

Suppose that:

$\forall M \in \R_{>0}: \exists N \in \R_{<0}: \forall x < N : \map f x > M$

for $M$ sufficiently large.

Then we write:

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

or

$\map f x \to +\infty$ as $x \to -\infty$

This is voiced:

$\map f x$ increases without bound as $x$ decreases without bound.

or:

$\map f x$ tends to (plus) infinity as $x$ tends to minus infinity.

### Decreasing Without Bound

Suppose that:

$\forall M \in \R_{<0}: \exists N \in \R_{<0}: x < N \implies \map f x < M$

for $M$ sufficiently large in magnitude.

Then we write:

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

or

$\map f x \to +\infty \ \text{as} \ x \to -\infty$

This is voiced:

$\map f x$ decreases without bound as $x$ decreases without bound.

or:

$\map f x$ tends to minus infinity as $x$ tends to minus infinity.