2 functions f and g that dont have limits at a number c but fg and f+g do

[ibensous]

I was wondering if anyone knows of an example where f and g are two functions that do not have limits at the real number c but f+g and fg have limits at c.

I know that if f and g are functions and L= lim_x->c f(x) and D = lim_x->c g(x) then the lim_x->c (f+g) = L + D and lim_x->c (fg) = LD but that's assuming both L and D exist. What if L and D don't exist?

 

[l'Hôpital]

What if f = -g = 1/x?

lim x--> 0 of f or g is undefined, but lim x--> 0 f + g = 0

lim x--> 0 fg would be - infinity.