Editing Big O notation (section)

== Example ==
In typical usage the <math>O </math> notation is asymptotical, that is, it refers to very large <math>x </math>.  In this setting, the contribution of the terms that grow "most quickly" will eventually make the other ones irrelevant. As a result, the following simplification rules can be applied:
*If <math>f(x) </math> is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted.
*If <math>f(x) </math> is a product of several factors, any constants (factors in the product that do not depend on <math>x </math>) can be omitted.
For example, let <math>f(x)=6x^4-2x^3+5 </math>, and suppose we wish to simplify this function, using <math>O </math> notation, to describe its growth rate as <math>x \rightarrow \infty </math>. This function is the sum of three terms: <math>6x^4 </math>, <math>-2x^3 </math>, and <math>5 </math>. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of <math>x </math>, namely <math>6x^4 </math>. Now one may apply the second rule: <math> 6x^4 </math>is a product of <math>6 </math> and <math>x^4 </math> in which the first factor does not depend on <math>x </math>. Omitting this factor results in the simplified form <math>x^4 </math>. Thus, we say that <math>f(x) </math> is a "big O" of <math>x^4 </math>. Mathematically, we can write <math>f(x)=O(x^4) </math>. One may confirm this calculation using the formal definition: let <math>f(x)=6x^4-2x^3+5 </math> and <math>g(x)=x^4 </math>. Applying the [[#Formal definition|formal definition]] from above, the statement that <math>f(x)=O(x^4) </math> is equivalent to its expansion,
<math display="block">|f(x)| \le  M x^4</math>
for some suitable choice of a real number <math>x_0 </math> and a positive real number <math> M </math> and for all <math>x>x_0 </math>. To prove this, let <math>x_0=1 </math> and <math>M=13 </math>. Then, for all <math>x>x_0 </math>:
<math display="block">\begin{align}
|6x^4 - 2x^3 + 5| &\le 6x^4 + |2x^3| + 5\\
                  &\le 6x^4 + 2x^4 + 5x^4\\
                  &= 13x^4
\end{align}</math>
so
<math display="block"> |6x^4 - 2x^3 + 5| \le 13 x^4 .</math>