To my very limited knowledge, division by 0 is undefined precisely because it breaks the field axioms. No dividing by 0 if you want a field. However there do exist structures that are not fields which allow for division by 0. Like the Riemann sphere. Or wheel algebras.
However a very common argument/"proof" I often hear for why division by 0 is undefined, is that $\lim_{x \to 0+} \frac{1}{x} = \infty$ whereas $\lim_{x \to 0-} \frac{1}{x} = -\infty$. And therefore since the limits go the opposite direction, this means we can't say $\frac{1}{0} = \infty$.
But I do not understand this line of reasoning. If the left and right limits differ, that would just mean the limit from both sides isn't defined. It wouldn't tell me anything about the function's value at 0, only near 0? And I don't see why a function's behaviour near 0 would have anything to do with it's value at 0.
Is there a missing step? Or is it just a fallacy?
Edit: A few people just agreeing with me haha, so let me try put it another way. Is there anything, anything at all, that we can conclude about division by 0, specifically from the differing left/right limits of $\frac{1}{x}$?