While I enjoy your overall tone and enthusiasm, there's no guarantee doing a weird operation will create a new useful mathematical object. The imaginary unit, i, follows associative/commutativity laws etc and is so enshrined. However, defining something like 1/0 as the "zero divided unit" it doesn't follow any of those laws to make it a group etc. So it can't be manipulated like a normal quantity in your algebra. Though it could be manipulated in other algebraic structures, such as wheels. However, if you haven't thought through this before, I think you totally have the right attitude, the right way you want to approach mathematics.
I don't really have the discipline to sit down and actually learn some useful math, however I enjoy the subject at a superficial level, so... that is, everything I say is wrong.
There was an interesting comment in the CouriusMarc video I linked. Quoted in full.
"Folks, no need to argue about colorful alternative mathematical theories - the thorny problem of division by zero was solved for good over 100 years ago by the rigorous development of infinitesimal calculus. Which says: division of a positive non-zero constant by something that tends to zero, tends to infinity [added note: dividing "zero by zero", or more exactly, two things that tend towards zero, is more complicated: it can give zero, infinity, or anything in-between, but that's for another time...]. So the calculator sort of gives the right answer, using almost the correct method: trying to fit an infinitesimally small number into a big one, and finding it fits so many times it goes to infinity. I would put it in the category of happy mechanical accidents."
Which fails to entirely satisfy, I suspect this is because it focuses on the mechanics of one view of the problem and leaves many other views still conflicting. The heart of the problem, that is, why the operation tends to be left as undefined, is that to be mathematically rigorous you have to satisfy all views as to what division actually means.
Yeah, the limit view is only defined in specific cases. For instance, if you have ((x+3)(x-5))/(x-5), this expression evaluates at x=5 to 0/0, but approaching it from either side, evaluates to 8. However, (x+3)/(x-5) at x=5 will be approaching -inf on one side and inf on the other.
I have an anecdote here too. When I learned about automatic differentiation, I very excitedly told one of my formalist math professors about the quantity e, e^2=0, e!=0, and how it could be used to compute the derivative of a function. He couldn't understand and essentially asked me to motivate the number and sketch the proofs that it's a well-behaved extension, which I couldn't do at all at the time, and he dismissed the whole thing which is actually the basis for some pretty important machine learning these days, among other things. So sometimes you can do pretty weird operations to get an extension, but without an intuitive grasp of how it works - either with a proof it is a well -behaved extension or by analogy for example, it's really hard to tell how useful such a concept is to a mathematician, how well it plays by the rules.
Being mathematically rigorous, you will have to define what the views you are talking about. In the one input function limit case, you need to check the limit from both sides. In the 2 input case, you need to check all possible approach paths to prove the limit exists. It gets out of hand. However, these are different views within a formal mathematical system. We don't need to reach consensus among different formal systems which happen to use the same symbols. For instance, I can define the set {cow,0} and equip it with operator '/' such that cow/0= and 0/cow=0 etc. This might be isomorphic to Z mod 2. Then cow by zero is defined, yet we can still agree that division by 0 is undefined, because when we are in general talking about division by 0, we're talking about an operation in a relatively specific context - how numbers behave. So, there really can be a very wide set of perspectives, and being rigorous must also specify the perspectives that count. Also - the cow case is pretty trivial, though this point could also be made with more complex structures, like wheels, that may also have useful physical interpretations, and they still wouldn't affect the collective agreement here because it's a contextually isolated case.
2) ignore (subtract away) the terms not proportional to dx
3) of the terms proportional, only care about terms minimally proportional (i.e. divide by dx, keep terms no longer proportional to dx, eliminate any still proportional to dx.)
The last step corresponds to eliminating terms proportional to dx^2 or higher
