If limx→af(x)=L and limx→ag(x)=m, then limx→a(f(x))g(x)=Lm
False
If both the limits limx→af(x) and limx→ag(x) exists finitely, then we can apply the limits independently to them in a limt involving both of them and x→a, provided the limit does not become indeterminate or infinite.
This is true for taking powers also, like f(x)g(x). We do not need the exact proof of this. A fair intuition is good enough to understand coming concepts and solve problems.