Find Order Notation of x√(1+x^2)

[coverband]

Teacher has:

x\sqrt{1+x^2}=x+\frac{1}{2}x^3...(= O(x))

in finding order notation of expression.

How is L.H.S. equal to R.H.S.?

 

[ystael]

The first equation is an expansion of the function x\sqrt{1 + x^2} in a Taylor series at zero, using the binomial series.

The second equation is a statement that the series expansion is bounded by a constant multiple of x. This is true only near zero, not near infinity (near infinity, \sqrt{1 + x^2} looks like |x|, so x\sqrt{1 + x^2} grows quadratically). Near zero, the written-out version of the big-O notation is: there exist constants C > 0 and \delta > 0 so that, whenever |x| < \delta, |x\sqrt{1 + x^2}| < C|x|. This statement is abbreviated x\sqrt{1 + x^2} = O(x) \textrm{ as } x \to 0.