Editing Asymptotic analysis (section)

{{short description|Description of limiting behavior of a function}}
{{about|the behavior of functions as inputs approach infinity or some other limit value|asymptotes in [[geometry]]|Asymptote}}

In [[mathematical analysis]], '''asymptotic analysis''', also known as '''asymptotics''', is a method of describing [[Limit (mathematics)|limiting]] behavior.

As an illustration, suppose that we are interested in the properties of a function {{math|''f''&hairsp;(''n'')}} as {{mvar|n}} becomes very large. If {{math|1=''f''(''n'') = ''n''<sup>2</sup> + 3''n''}}, then as {{mvar|n}} becomes very large, the term {{math|3''n''}} becomes insignificant compared to {{math|''n''<sup>2</sup>}}. The function {{math|''f''(''n'')}} is said to be "''asymptotically equivalent'' to {{math|''n''<sup>2</sup>}}, as {{math|''n'' → ∞}}". This is often written symbolically as {{math|''f''&hairsp;(''n'') ~ ''n''<sup>2</sup>}}, which is read as "{{math|''f''(''n'')}} is asymptotic to {{math|''n''<sup>2</sup>}}".

An example of an important asymptotic result is the [[prime&nbsp;number theorem]]. Let {{math|π(''x'')}} denote the [[prime-counting function]] (which is not directly related to the constant [[pi]]), i.e. {{math|π(''x'')}} is the number of [[prime number]]s that are less than or equal to {{mvar|x}}. Then the theorem states that
<math display="block">\pi(x)\sim\frac{x}{\ln x}.</math>

Asymptotic analysis is commonly used in [[computer science]] as part of the [[analysis of algorithms]] and is often expressed there in terms of [[big O notation]].