Take
f(x)=1 for x<=c and f(x)=-1 for x>c,
g(x)=1 for x>c and g(x)=-1 for x<=c.
I took:
-the function f that is 1 if x is less than or equal to c and is -1 if x is greater than c
-the function g that is -1 if x is less than or equal to c and is 1 if x is greater than c
Both functions don't have a limit at c (approaching c from the left gives a different value than approaching c from the right), also you have f+g = 0 and f*g= -1 so that both f+g and f*g have limits at c.