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

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$f(x_0)$ $g(x_0)$ $f'(x_0)$ $g'(x_0)$ Conclusion about $(f \cdot g)'(x_0)$ Explanation
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