I was thinking perhaps the limit of a function at a constant or at infinity is not unique. Thus by delta-epsilon I mean there exists another number L' such that satisfies the condition of the limit. Thus, prove that if the limit exists it is unique. I was able to show that if L is one limit and L' is another limit then (L+L')/2 is also a limit. Thus, this shows that if a limit is not unique there exists infinitely many limits for that function. But this is not true I was trying to proof that because my books on advanced calculus did not consider this concept.