Definition:Limit of Real Function/Limit at Infinity/Positive
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Definition
Let $f$ be a real function defined on an open interval $\openint a \to$.
Let $L \in \R$.
$L$ is the limit of $f$ at 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 $x \to +\infty$
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.
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
- $\map f x \to -\infty$ 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
- 1970: N.G. de Bruijn: Asymptotic Methods in Analysis (3rd ed.) ... (previous) ... (next): $1.1$ What is asymptotics? $(1.1.3)$, $(1.1.4)$