Asymptotic limit

In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The method has applications across science. Examples are:

A simple illustration, when considering a function f(n), is when there is a need to describe its properties as n becomes very large. Thus, if f(n) = n²+3n, the term 3n becomes insignificant compared to n², when n is very large. The function f(n) is said to be "asymptotically equivalent to n² as n → ∞", and this is written symbolically as f(n) ~ n².

Formally, given functions f and g of a natural number variable n, one defines a binary relation

if and only if (according to Erdelyi, 1956)

This relation is an equivalence relation on the set of functions of n. The equivalence class of f informally consists of all functions g which are approximately equal to f in a relative sense, in the limit.

If ${\displaystyle f\sim g}$, then

${\displaystyle f^{r}\sim g^{r}}$for any real r, and

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