Using the product rule for differentiation for limiting behavior at points with undefined derivative
From Calculus
Summary
Conclusion about | Explanation | ||||
---|---|---|---|---|---|
finite | finite | undefined | undefined | insufficient information (could be finite or undefined) | We don't know the details behind the undefined |
nonzero | nonzero and same sign as | vertical tangent | vertical tangent of same type as for (i.e., either both are increasing or both are decreasing) | vertical tangent, type (increasing/decreasing) is determined by signs of and types of vertical tangent for | [SHOW MORE] |
nonzero | nonzero and opposite sign to | vertical tangent | vertical tangent of same type as for (i.e., either both are increasing or both are decreasing) | insufficient information | [SHOW MORE] |
nonzero | nonzero and same sign as | vertical tangent | vertical tangent of opposite type as for (i.e., one is increasing and one is decreasing) | insufficient information | |
nonzero | nonzero and opposite sign to | vertical tangent | vertical tangent of opposite type as for (i.e., one is increasing and one is decreasing) | vertical tangent, type depends on signs | |
zero | known whether it is zero, positive, or negative | known whether it is finite, vertical tangent, etc. | vertical tangent | insufficient information in all cases. |