Definition:Limit of Real Function/Limit at Infinity/Negative
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Definition
Let $f$ be a real function defined on an open interval $\openint \gets a$.
Let $L \in \R$.
$L$ is the limit of $f$ at minus infinity if and only if:
- $\forall \epsilon \in \R_{>0}: \exists c \in \R: \forall x < c: \size {\map f x - L} < \epsilon$
This is denoted as:
- $\ds \lim_{x \mathop \to - \infty} \map f x = L$
Increasing Without Bound
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.
Decreasing Without Bound
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
- $\map f x \to +\infty \ \text{as} \ x \to -\infty$
This is voiced:
- $\map f x$ decreases without bound as $x$ decreases without bound.
or:
- $\map f x$ tends to minus 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