Difference between revisions of "Asymptotic equivalence"
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'''Asymptotic equivalence''' is a notion of [[function]]s "eventually" becoming "essentially equal". | '''Asymptotic equivalence''' is a notion of [[function]]s "eventually" becoming "essentially equal". | ||
| − | More precisely, let <math>f</math> and <math>g</math> be functions of a [[real number | real]] variable. We say that <math>f</math> and <math>g</math> are '''asymptotically equivalent''' if the [[limit]] <math>\lim_{x\to \infty} \frac{f(x)}{g(x)}</math> exists and is equal to 1. | + | More precisely, let <math>f</math> and <math>g</math> be functions of a [[real number | real]] variable. We say that <math>f</math> and <math>g</math> are '''asymptotically equivalent''' if the [[limit]] <math>\lim_{x\to \infty} \frac{f(x)}{g(x)}</math> exists and is equal to 1. We sometimes denote this as <math>f \sim g</math>. |
| − | Let us consider functions of a common [[domain (function) | domain]] that are nonzero for sufficiently large arguments. Evidently, | + | Let us consider functions of a common [[domain (function) | domain]] that are nonzero for sufficiently large arguments. Evidently, all such functions are asymptotically equivalent to themselves, and if <math>f \sim g</math>, then |
<cmath> \lim_{x\to \infty} \frac{g(x)}{f(x)} = \frac{1}{\lim_{x\to \infty} f(x)/g(x)} = 1 , </cmath> | <cmath> \lim_{x\to \infty} \frac{g(x)}{f(x)} = \frac{1}{\lim_{x\to \infty} f(x)/g(x)} = 1 , </cmath> | ||
so <math>g \sim f</math>. Finally, it is evident that if <math>f \sim g</math> and <math>g\sim h</math>, then <math>f \sim h</math>. Asymptotic equivalence is thus an equivalence relation in this context. | so <math>g \sim f</math>. Finally, it is evident that if <math>f \sim g</math> and <math>g\sim h</math>, then <math>f \sim h</math>. Asymptotic equivalence is thus an equivalence relation in this context. | ||
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The functions <math>f(x) = x^2</math> and <math>g(x) = x^2 + x</math> are asymptotically equivalent, since | The functions <math>f(x) = x^2</math> and <math>g(x) = x^2 + x</math> are asymptotically equivalent, since | ||
| − | <cmath> \lim_{x\to infty} \frac{f(x)}{g(x)} = \lim_{x\to\infty} \left( 1 - \frac{1}{x | + | <cmath> \lim_{x\to \infty} \frac{f(x)}{g(x)} = \lim_{x\to\infty} \left( 1 - \frac{1}{x+ 1} \right) = 1 . </cmath> |
On the other hand the functions <math>f(x) = x^2</math> and <math>g(x) = x^3</math> 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. | On the other hand the functions <math>f(x) = x^2</math> and <math>g(x) = x^3</math> 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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Latest revision as of 13:15, 30 March 2014
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
![\[\lim_{x\to \infty} \frac{g(x)}{f(x)} = \frac{1}{\lim_{x\to \infty} f(x)/g(x)} = 1 ,\]](http://latex.artofproblemsolving.com/4/3/e/43eb839236182a470c9dd29cb8c11a7a27c15932.png)
so 
. Finally, it is evident that if and 
, then 
. Asymptotic equivalence is thus an equivalence relation in this context.
Examples
The functions 
and 
are asymptotically equivalent, since
![\[\lim_{x\to \infty} \frac{f(x)}{g(x)} = \lim_{x\to\infty} \left( 1 - \frac{1}{x+ 1} \right) = 1 .\]](http://latex.artofproblemsolving.com/b/3/a/b3a649ac066a47750a33771bd179d100de397b69.png)
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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