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

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


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

That is, $-\map f x$ can be made arbitrarily large by making $x$ sufficiently large.

This is voiced:


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

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

Also see

 * Definition:Unbounded Mapping
 * Definition:Unbounded Divergent Sequence


 * Definition:Infinite Limit at Infinity