English: Graph showing a case where Newton's method fails to converge on a root. Newton's method applied to the function
${\displaystyle f\left(x\right)={\begin{cases}{\sqrt {x-4}},&{\mbox{for}}\,\,x\geq 4\\-{\sqrt {4-x}},&{\mbox{for}}\,\,x<4\end{cases}}}$
starting with ${\displaystyle x_{0}=2}$. For this function choosing any ${\displaystyle x_{1}=r-h}$ then ${\displaystyle x_{2}=r+h}$ would cause successive approximations to alternate back and forth, so no amount of iteration would get us any closer to the root than our first guess. Function has inflection point ( method has a cycle and never converge )