Category:Little-O Notation

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This category contains results about little-$\oo$ notation.
Definitions specific to this category can be found in Definitions/Little-O Notation.

Sequences

Let $g: \N \to \R$ be a real sequence, expressed here as a real-valued function on the set of natural numbers $\N$.


Then $\map \oo g$ is defined as:

$\map \oo g = \set {f: \N \to \R: \forall c \in \R_{>0}: \exists n_0 \in \N: \forall n > n_0: \size {\map f n} \le c \cdot \size {\map g n} }$


Estimate at Infinity

Let $f$ and $g$ be real functions defined on a neighborhood of $+\infty$ in $\R$.

Let $\map g x \ne 0$ for $x$ sufficiently large.


$f$ is little-$\oo$ of $g$ as $x \to \infty$ if and only if:

$\ds \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = 0$

Subcategories

This category has only the following subcategory.