Asymptotics

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function f(n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n2. The function f(n) is said to be "asymptotically equivalent to n2, as n → ∞". This is often written symbolically as f(n) ~ n2, which is read as "f(n) is asymptotic to n2".
An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less than or equal to x. Then the theorem states that




π
(
x
)
∼


x

ln
⁡
x



.


{\displaystyle \pi (x)\sim {\frac {x}{\ln x}}.}

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