Yes and no. In exact terms, the dividend asymptotically approaches infinity (either positive or negative) as the denominator approaches zero. Asymptotically means it "never quite gets there".
In more concrete (and practical) terms, the answer is infinity, or at least a number large enough to respresent a ridiculously large value (which can still be positive or negative).
The issue comes up in computing because computers can only handle certain ranges of numbers, with the end points of those ranges determined by the size of the container the value is stored in. The programmer can protect against overflow by preemtively testing for too small values in the denominator and substituting a very large number in place of the actual calculation, or they can reactively catch the overflow error (using exception handling) and substitute afterward. Or, the programmer can allow the built-in exception handling to take care of the problem, which commonly causes the program to stop working altogether.
I have to admit, though, "nullity" is exactly the WRONG word for this value continuum, because "null" is nothing, and therefore "nullity" cannot be defined rationally to mean "anything".
BTW - If I have insulted your intelligence or experience with my explanation, I apologize. You sounded as though you might appreciate a fairly complete answer.
Ah ha, someone else who rejects the name. I think it should be called a pixie...
The problem with thinking about it as "approaching infinity" is that it implies that, as n approaches infinity, 0*n approaches every number.
Because if X/0 approaches infinity, then 0*infinity would be X, where X is any number.
But in fact 0*n as n approaches infinity is ZERO.
Which is why we don't usually say that X/0=> infinity, but rather than you can't divide X by zero.
I am a programmer with a fairly extensive mathematical background. However, the schooling has had a fair number of years (not yet approaching infinity) to wear off. Your answer did not at all offend me.