I wasn't looking to get in an argument, and knew even replying at all was likely a mistake, but went against my instincts and did it, for reasons unknown to me (I'd just awoken from a short nap, so maybe I was still a bit foggy). I'll cut to the chase and share what I think is a cleaner argument. If we limit our attention to just algebra, and fields in this case, the reason division by zero is undefined is that division in fields is just shorthand for multiplicative inverses (i.e., inverse of A is any element, B, which multiplied times A gives the multiplicative identity, 1). 0 x B = 0 always, it's "easy" to show using the axioms, thus there is no multiplicative inverse for 0 (I guess you might also need to show 0 <> 1). But many people don't really learn about number systems this way, and the "easy" part I mentioned above may not be so obvious (comes down to using the distributive property and axioms about 0). There's no need to go to limits or sequences, in other words.