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 $$x\sqrt{1 + x^2} = O(x) \textrm{ as } x \to 0$$.