Definition:Little-O Notation/Real
< Definition:Little-O Notation(Redirected from Definition:Little-O Estimate for Real Function)
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Definition
Estimate at infinity
Let $\map g x \ne 0$ for $x$ sufficiently large.
$f$ is little-$\oo$ of $g$ as $x \to \infty$ if and only if:
- $\ds \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = 0$
Point Estimate
Definition:Little-O Notation/Real/Point
Also known as
The little-$\oo$ notation, along with big-$\OO$ notation, are also referred to as Landau's symbols or the Landau symbols, for Edmund Georg Hermann Landau.
Examples
Example: Sine Function at $+\infty$
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = \sin x$
Then:
- $\map f x = \map \oo x$
as $x \to \infty$.
Example: $x = \map \oo {x^2}$ at $+\infty$
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = x$
Then:
- $\map f x = \map \oo {x^2}$
as $x \to \infty$.