Definition:Limit of Real Function/Limit at Infinity/Negative/Decreasing Without Bound

Definition

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

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.

Also see

• Definition:Unbounded Mapping
• Definition:Unbounded Divergent Sequence

Sources

• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 3.5$