Ked

I think what you've missed is that infinity isn't really a number, it's just a symbol which means an undefined result. The only time we distinguish between +ve and -ve infinity is when we use the expression "The limit of F(x) as x tends to infinity" or "The limit of F(x) as x tends to - infinity"


But strictly speaking these expressions are mathematical slang, what we mean by "The limit of F(x) as x tends to infinity is q" is

For any real number d > 0 there exists a real number Y such that if a real number X > Y then |F(x) - q| < d


For calculating limits of functions such as Sin and Cos we may have to redefine this slightly but we still avoid the mention of +ve and -ve infinity in our definition.

When you think of the left and right limits of 1/0 consider this

1/0 = exp(ln(1/0)) = exp(-ln(0)) = exp(-(-inf)) = +ve infinity


This is technicly a proof by contradiction, (to stop your argument that it doesn't count because I've used +ve and -ve infinity to proove they don't exist) If the 2 infinities are distinct then it can be shown that they are indistinct and hence there is a contradiction, hence they cannot be distinct.

The reason that +ve and -ve infinity are used in the notation of limits is that it's far easier than writing down the formal definition of what you are doing, Everyone understands the notation Lim(x-> -inf) F(x) If they want a formal definition they can go to Cambridge, Harvard, The Royal Society, or wherever the argument over the precice definition happens to be taking place and get a big page full of symbols that define it. +/- infinity aren't numbers, they're ways of defining 2 particular properties of a function