 Solution

The Newton root-finding iteration isIt requires $f$ and its derivative to be defined at the current point, and requires $f^{\prime }(x)\ne 0$. A step can leave the domain, making a subsequent iteration undefined. In floating-point arithmetic, division by zero, overflow or non-finite function evaluations can also prevent a usable step. At a root where $f^{\prime }$ is zero the displayed quotient itself remains undefined unless the implementation stops before evaluating it.