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

## Definition

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

Suppose that:

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

for $M$ sufficiently large.

Then we write:

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

or

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

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

This is voiced:

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

or:

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