Can we make illegitimate operations with $0$ legitimate by adding a few more axioms? Just out of curiosity, can we make illegitimate operations with $0$, say, division by $0$ legitimate simply by imposing additional axioms? If so, then what consequences may follow? If the answer is negative, what are the differences between making operations with $+\infty, -\infty$ legitimate in the extended reals?

In some contexts, things like $5/0$ should be taken to be $\infty$, where that is neither $+\infty$ nor $-\infty$, but a single $\infty$ at both ends of the real line or at the extreme of the complex plane.  This makes all trigonometric functions and rational functions continuous.
This is really an example of one of the points at infinity in projective geometry.
