Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 19 / b / i

The positive polynomial root is $x_{\ast }=\epsilon +\sqrt{1+{\epsilon }^2}$. Its Taylor expansion near zero isThe starting error is therefore $x_0-x_{\ast }=-\epsilon -\frac{1}{2}{\epsilon }^2+O({\epsilon }^4)$. ConsequentlyFor signed $\epsilon$, the magnitude scaling should be written with $|\epsilon |$; this is the precise interpretation of the proportionality in the question.