the division operation is just not defined when the denominator is 0. it’s just that simple. no way to justify how it is defined is valid. limits are not valid, because taking the limit is not the same operation as division.

most operators we use these days, including multiplication, division, addition, and subtraction, have their definitions in group and ring theory (abstract algebra).

an operator is just a function. like all functions, it needs a definition. the division operator we most often talk about is defined from an ordered pair of reals to a real. if the second of the pair it operates on is 0, there is no number it yields. it’s just defined that way.

asking what a number divided by 0 yields is like asking what radiohead marrying a prime number yields. it’s just nonsensical. the words and operators just aren’t defined to be meaningful with those terms.