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I am having difficulty getting to grips with “the order notation”. I have the following definition…

“A function f(x) is said to be of order x^n as x -> 0 if there is a non-zero constant C such that |f(x)|<C|x^n| for all x in an interval around x=0. This is written as

f(x) = O(x^n) as x->0”

some examples:

x(1+x^2)^1/2 = x + x^3/2 +… = O(x)

x/(1+x) = x(1 – x + x^2 +…) = O(x)

(x+b)^a – x^a = x^a(1 + ab/x +…) –x^a = O[x^(a-1)]

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# Order notation: f(x)=O(x^n) as x-> 0

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