Definition:Limit of Real Function/Limit at Infinity/Positive/Decreasing Without Bound
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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:
- $\ds \lim_{x \mathop \to +\infty} \map f x = -\infty$
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.
Also see
Sources
- 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 3.5$