\begin{definition}[Definition:Limit of Real Function/Limit at Infinity/Positive/Decreasing Without Bound]
Let $f$ be a real function defined on an open interval $\openint a \to$.
Suppose that:
:$\forall M \in \R_{<0}: \exists N \in \R_{>0}: x > N \implies \map f x < M$
for $M$ sufficiently large in magnitude.
Then we write:
:$\ds \lim_{x \mathop \to +\infty} \map f x = -\infty$
or
:$\map f x \to -\infty$ 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'''.
\end{definition}