What do we mean when we say a function is asymptotic?

What do we mean when we say a function is asymptotic?

Two functions $f(x)$ and $g(x)$ are said to be asymptotic if a binary relation is defined on them as thus:

$f(x)\sim{g(x)}$ as $x\rightarrow\infty$ if and only if $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1$ (de bruijn, 1981).

In this case the functions are said to be asymptotically equivalent because the symbol $\sim$ defines an equivalence relation on the set of functions of $x$.

The domain of $f$ and $g$ can be defined over any set of numbers e.g $\mathbb{R},\mathbb{C},\mathbb{Z}^+$. 

The study of asymptotes seeks to descrobe the limit behaviour of functions. 

Example of a simple asymptotic functions can be obtained in the function $f(n)=n^2+3n$, as $n$ becomes very large the  term $3n$ becomes small and insignificant compared to $n^2$. The function $f(n)$ is said to be "asymptotically equivalent to $n^2$, as $n\rightarrow\infty$". This is often written symbolically as $f(n)\sim{n}^2$, which is read as "f(n) is asymptotic to n2".

An example of a special case of an asymptotic relation is the Stirling approximation $n!\sim\sqrt{2\pi{n}}(\frac{n}{e})^n$. 

Properties of asymptotic functions

If $f\sim{g}$ and $a\sim{b}$, then the following holds:

1. $f\times{a}\sim{g}\times{b}$
2.  $f/a\sim{g/b}$
3.  $\log(f)\sim\log(g)$
4. $f^r\sim{g^r}$ for every real $r$

These properties allow asymptotically-equivalent functions to be freely exchanged in many algebraic expressions. 

 

Read more on asymptote. Wikipedia: Asymptotic analysis