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Asymptotic equivalence

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Asymptotic equivalence is a notion of functions "eventually" becoming "essentially equal".

More precisely, let f and g be functions of a real variable. We say that f and g are asymptotically equivalent if the limit \lim_{x\to \infty} \frac{f(x)}{g(x)} exists and is equal to 1. We sometimes denote this as f \sim g.

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 f \sim g, then \lim_{x\to \infty} \frac{g(x)}{f(x)} = \frac{1}{\lim_{x\to \infty} f(x)/g(x)} = 1 , so g \sim f. Finally, it is evident that if f \sim g and g\sim h, then f \sim h. Asymptotic equivalence is thus an equivalence relation in this context.

Examples

The functions f(x) = x^2 and g(x) = x^2 + x are asymptotically equivalent, since \lim_{x\to infty} \frac{f(x)}{g(x)} = \lim_{x\to\infty} \left( 1 - \frac{1}{x^2 + x} \right) = 1 . On the other hand the functions f(x) = x^2 and g(x) = x^3 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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