\begin{definition}[Definition:Limit of Real Function/Limit at Infinity/Positive/Increasing Without Bound]
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'''.
\end{definition}