Editing Asymptotic analysis (section)

== Definition ==
Formally, given functions {{math|''f''&hairsp;(''x'')}} and {{math|''g''(''x'')}}, we define a binary relation
<math display="block">f(x) \sim g(x) \quad (\text{as } x\to\infty)</math>
if and only if {{Harv|de&nbsp;Bruijn| 1981| loc= §1.4}}
<math display="block">\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1.</math>

The symbol {{math|~}} is the [[tilde]]. The relation is an [[equivalence relation]] on the set of functions of {{mvar|x}}; the functions {{mvar|f}} and {{mvar|g}} are said to be ''asymptotically equivalent''. The [[Domain of a function|domain]] of {{mvar|f}} and {{mvar|g}} can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.

The same notation is also used for other ways of passing to a limit: e.g. {{math|''x'' → 0}}, {{math|''x'' ↓ 0}}, {{math|{{abs|''x''}} → 0}}. The way of passing to the limit is often not stated explicitly, if it is clear from the context.

Although the above definition is common in the literature, it is problematic if {{math|''g''(''x'')}} is zero infinitely often as {{mvar|x}} goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in [[little-o notation]], is that {{math|''f'' ~ ''g''}} if and only if
<math display="block">f(x)=g(x)(1+o(1)).</math>

This definition is equivalent to the prior definition if {{math|''g''(''x'')}} is not zero in some [[Neighbourhood (mathematics)|neighbourhood]] of the limiting value.<ref>{{SpringerEOM |id=Asymptotic_equality| title=Asymptotic equality}}</ref><ref>{{Harvtxt|Estrada|Kanwal|2002| loc=§1.2}}</ref>