This website does readability filtering of other pages. All styles, scripts, forms and ads are stripped. If you want your website excluded or have other feedback, use this form.

Asymptotic equality - Encyclopedia of Mathematics The European Mathematical Society


      Asymptotic equality

      From Encyclopedia of Mathematics Jump to: navigation, search

      Two functions $f(x)$ and $g(x)$ are called asymptotically equal as $x\to x_0$ if in some neighbourhood of the point $x_0$ (except possibly at $x_0$ itself)



      $$\lim_{x\to x_0}\epsilon(x)=1,$$



      as $x\to x_0$ ($x_0$ is a finite or an infinite point of the set on which the functions under consideration are defined). If $g(x)$ does not vanish in some neighbourhood of $x_0$, this condition is equivalent to the requirement

      $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1.$$

      In other words, asymptotic equality of two functions $f(x)$ and $g(x)$ as $x\to x_0$ means, in this case, that the relative error of the approximate equality of $f(x)$ and $g(x)$, i.e. the magnitude $[f(x)-g(x)]/g(x)$, $g(x)\neq0$, is infinitely small as $x\to x_0$. Asymptotic equality of functions is meaningful for infinitely-small and infinitely-large functions. Asymptotic equality of two functions $f(x)$ and $g(x)$ is denoted by $f(x)\sim g(x)$ as $x\to x_0$, and is reflexive, symmetric and transitive. Accordingly, the set of infinitely-small (infinitely-large) functions as $x\to x_0$ is decomposed into equivalence classes of such functions. An example of asymptotically-equal functions (which are also called equivalent functions) as $x\to x_0$ are the functions $u(x)$, $\sin u(x)$, $\ln[1+u(x)]$, $e^{u(x)}-1$, where $\lim_{x\to x_0}u(x)=0$.

      If $f\sim f_1$ and $g\sim g_1$ as $x\to x_0$, then

      $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=\lim_{x\to x_0}\frac{f_1(x)}{g_1(x)},$$

      where the existence of any one of the limits follows from the existence of the other one. See also Asymptotic expansion of a function; Asymptotic formula.


      One also says that $f(x)$ and $g(x)$ are of the same order of magnitude at $x_0$ instead of asymptotically equal.


      [a1] R. Courant, "Differential and integral calculus" , 1 , Blackie (1948) pp. Chapt. 3, Sect. 9 (Translated from German) How to Cite This Entry:
      Asymptotic equality. Encyclopedia of Mathematics. URL: This article was adapted from an original article by M.I. Shabunin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article Retrieved from "[]" Category:
      • This page was last modified on 7 July 2014, at 20:10.
      Manage Cookies