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[Jelle]

It's undefined, what more is there to it?
Just like there's (I'm using google translate here, don't know the english words) axioms for more broad cases and specific cases, sometimes there's a lack of one for a specific case. Because there's none for the case of m/n where n=0 it's not part of the mathematical structure. If you were to consider the same operation outside of the real numbers, including infinity it is defined to be equal to infinity, but with real numbers it simply has no meaning.

It's been quite a time since I did any real math, so don't hate me if I said something terrible false or used wrong wording. 

[Kay12]

Then again, if x/0 = ∞ and -0 = 0... does this mean that *gasp* ∞ = -∞?

[Virex]

Then again, if x/0 = ∞ and -0 = 0... does this mean that *gasp* ∞ = -∞?

In the projective extended real numbers, yes. but not in the affinely extended real numbers, which is the set that's usually used when dealing with limits.