Conditions for the failure of newtons method :

(a).

If  the  initial estimate or estimates are taken at a point at which there is a horizontal tangent line, then this line will never hit the -axis, and Newtons Method will fail to locate a root.

(b)

If there is a horizontal tangent line then the derivative is zero, and we cannot divide by  as the formula requires.

(c).

If the estimates oscillate back and forth then Newtons method will not work.

(d).

If there are two roots, we must have a first guess near the root that we are interested in, otherwise Newtons method will find the wrong root.

(e).

The derivative may be zero at the root; the function may fail to be continuously differentiable and we may choose a wrong estimate  that lies outside the range of guaranteed convergence.

(f).

Finally, If there are no roots, then Newtons method will fail to find it. 

The Conditions for the failure of newtons method are mentioned.