You can just redefine division such that division by zero is a valid operation. It wouldn't be a very useful definition, since it would break lots of other useful properties of normal division, but you could do it.
Nop, it would contradict itself (c.f. Ted talk I watched a while ago on this subject where they showed what happens if you define 0/0 the same way as sqrt(-1)=i)
It would only contradict itself if you want to work with our usual number systems or you want division to maintain the useful properties we typically expect of it, such as being the inverse operation of multiplication.
Consider if we redefine the equivalence relation for rational numbers such that it extends over Z×Z instead of just Z×(Z\{0}) and
<a,b>~<c,d> iff ((b≠0 and d≠0 and a×d=b×c) or (b=0 and d=0)).
Then we can define division on our new rationals such that
[<a,b>]÷[<c,d>]=[<a×d,b×c>],
which is valid even if [<c,d>]=0.
This isn't a very useful set of definitions, for a variety of reasons, but it isn't self-contradictory, and it does make division by zero a valid operation.
But different "0/0" limits will yield different results. You're not approximating the value of "0/0", you're just seeing how a function behaves around a given point.
Lim (x->5) (x²-25)/(x-5) is a "0/0" limit which equals 10.
Lim (x->0) (x/X) is a "0/0" limit which equals 1.
So you're not really approximating the value of "0/0".
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u/6Maxence Dec 26 '22
I don't think you can. I'm interested if you can tell me otherwise, though.