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

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This discussion explores the behavior of two functions, f and g, that do not have limits at a specific point c, yet their sum (f + g) and product (fg) do exhibit limits. The example provided involves the functions f(x) = 1/x and g(x) = -1/x as x approaches 0. While both functions approach undefined limits as x approaches 0, their sum approaches 0, and their product approaches negative infinity, demonstrating that limits can exist for combined functions even when individual limits do not.

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

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