Definition:Asymptotic Equality

Definition

Sequences

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

$\sequence {a_n}$ is asymptotically equal to $\sequence {b_n}$ if and only if:

$\ds \lim_{n \mathop \to \infty} \dfrac {a_n} {b_n} = 1$

Real Functions

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

Then:

$f$ is asymptotically equal to $g$

if and only if:

$\dfrac {\map f x} {\map g x} \to 1$ as $x \to +\infty$.

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

General Definition

Let $T = \struct {S, \tau}$ be a topological space.

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

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:

$f - g = \map \oo g$ as $x \to x_0$

where $\oo$ denotes little-$\oo$ 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 equality can be found here.

Sources

• 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): Preface to first edition: Prerequisites