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

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] We consider the case where all signs are $+$. In this case, we get $(\to +)\cdot (\to +\infty )+(\to +\infty )\cdot (\to +)=\to +\infty$. The other cases work similarly.
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] For instance, if $f(x_{0})>0>g(x_{0})$, and the limits for $f'(x_{0})$ and $g'(x_{0})$ are both $+\infty$, then we get $(\to +)(\to +\infty )+(\to +)(\to -\infty )=(\to +\infty )+(\to -\infty )$. This is an indeterminate form, and the actual outcome depends on which of the approaches happens faster.
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.