Definition:Asymptotically Equal

From ProofWiki
Jump to navigation Jump to search

Definition

Sequences

Let $b_n \ne 0$ for all $n$.


$\left \langle {a_n} \right \rangle$ is asymptotically equal to $\left \langle {b_n} \right \rangle$ if and only if:

$\displaystyle \lim_{n\to\infty}\dfrac {a_n} {b_n} \to 1$


Functions

Let $f$ and $g$ real functions defined on $\R$.

Then:

$f$ is asymptotically equal to $g$

if and only if:

$\dfrac {f \left({x}\right)} {g \left({x}\right)} \to 1$ as $x \to +\infty$.

That is, the larger the $x$, the closer $f$ gets (relatively) to $g$.


General Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\, \cdot \,}\right\Vert$.

Let $f, g: S \to V$ be mappings.

Let $x_0 \in X$.


Then:

$f$ is asymptotically equal to $g$ as $x \to x_0$

if and only if:

$f - g = o \left({g}\right)$ as $x \to x_0$

where $o$ denotes little-O notation.


Notation

The notation $a_n \sim b_n$ and $f \sim g$ is frequently seen to denote asymptotic equality.


Also known as

If $f \sim g$, then $f$ and $g$ are also said to be asymptotically equivalent.


Also see

  • Results about Asymptotic Notation can be found here.