Asymptotic equivalence is a notion of functions "eventually" becoming "essentially equal".
More precisely, let and be functions of a real variable. We say that and are asymptotically equivalent if the limit exists and is equal to 1. We sometimes denote this as .
Let us consider functions of a common domain that are nonzero for sufficiently large arguments. Evidently, all such functions are asymptotically equivalent to themselves, and if , then so . Finally, it is evident that if and , then . Asymptotic equivalence is thus an equivalence relation in this context.
The functions and are asymptotically equivalent, since On the other hand the functions and are not asymptotically equivalent. In general, two real polynomial functions are asymptotically equivalent if and only if they have the same degree and the same leading coeffcient.
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