Definition:Limit of Real Function/Limit at Infinity/Negative/Increasing Without Bound
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Definition
Let $f$ be a real function defined on an open interval $\openint \gets a$.
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 $x \to -\infty$
This is voiced:
- $\map f x$ increases without bound as $x$ decreases without bound.
or:
- $\map f x$ tends to (plus) infinity as $x$ tends to minus 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.): $\S 3.5$