Using the product rule for differentiation for limiting behavior at points with undefined derivative

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Summary

f(x_0) g(x_0) f'(x_0) g'(x_0) Conclusion about (f \cdot g)'(x_0) 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 f(x_0) vertical tangent vertical tangent of same type as for f (i.e., either both are increasing or both are decreasing) vertical tangent, type (increasing/decreasing) is determined by signs of f,g and types of vertical tangent for f,g [SHOW MORE]
nonzero nonzero and opposite sign to f(x_0) vertical tangent vertical tangent of same type as for f (i.e., either both are increasing or both are decreasing) insufficient information [SHOW MORE]
nonzero nonzero and same sign as f(x_0) vertical tangent vertical tangent of opposite type as for f (i.e., one is increasing and one is decreasing) insufficient information
nonzero nonzero and opposite sign to f(x_0) vertical tangent vertical tangent of opposite type as for f (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.