1. Are differentials on the test? I have no clue what they are.
2. Do we get the newton's method equation? I don't feel like memorizing it.
3. What prevents Newton's method from happening?
4. What prevents a linearization from happening, such as what this question is asking?
1) No. Thanks for paying attention.
2) No. Stop whining.
3) Laziness, usually. Other than that, there could be continuity or differentiability issues. Furthermore, if the derivative at one of the x-values works out to 0, you have a problem with the tangent intersecting the x-axis (and an issue with division by 0 in the formula). There's also the "infinite loop" problem, shown in Figure 1 below:
4) Continuity or differentiability issues, including cusps, corners, asymptotes, holes, vertical tangents, blah, blah, blah. If there's an issue with finding the tangent slope somewhere, there's an issue with finding the linearization there too (which is really just the tangent line's equation).