Asymptotic equality

Two functions and are called asymptotically equal as if in some neighbourhood of the point (except possibly at itself)

where

i.e.

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

In other words, asymptotic equality of two functions and as means, in this case, that the relative error of the approximate equality of and , i.e. the magnitude , , is infinitely small as . Asymptotic equality of functions is meaningful for infinitely-small and infinitely-large functions. Asymptotic equality of two functions and is denoted by as , and is reflexive, symmetric and transitive. Accordingly, the set of infinitely-small (infinitely-large) functions as is decomposed into equivalence classes of such functions. An example of asymptotically-equal functions (which are also called equivalent functions) as are the functions , , , , where .

If and as , then

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 and are of the same order of magnitude at instead of asymptotically equal.

References

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