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.


The functions $f(x) = x^2$ and $g(x) = x^2 + x$ are asymptotically equivalent, since \[\lim_{x\to \infty} \frac{f(x)}{g(x)} = \lim_{x\to\infty} \left( 1 - \frac{1}{x+ 1} \right) = 1 .\] 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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