Asymptotic equality

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)

$$f(x)=\epsilon(x)g(x),$$

where

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

i.e.

$$f(x)=g(x)[1+o(1)],$$

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.

Comments

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

References

[a1]	R. Courant, "Differential and integral calculus" , 1 , Blackie (1948) pp. Chapt. 3, Sect. 9 (Translated from German)