But then you have to decide whether infinity has a sign; hence the extended real number line (which has +inf and -inf) versus the real projective number line (which just has an inf which is both positive and negative and neither at the same time). The extended real number line sort of doesn't make sense because the sign of the result you get in taking the limit varies depending upon which side you approach it from. However, the definitions of +/-inf are very useful in calculus and suchlike, and so it is commonly used.
IEEE754 (the standard for floating point numbers) defines:
|x|/0 = infinity
-|x|/0 = -infinity
And 0/0 is truly undefined: IEEE754 gives 0/0 = NaN (not a number).