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:


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

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