What point c were you thinking of for your choice of f(x) and g(x)?

How about f(x) = 0 if x is rational; 1 if x is irrational. g(x) = 1 if x is rational; 0 if x is irrational. Then f(x) + g(x) = 1, f(x)g(x) = 0, and the limit does not exist for f(x) or for g(x) at any point c.