0/0 is referred to as an
indeterminate form, along with 0*
infinity, infinity/infinity, infinity^0, 0^infinity, 1^infinity, infinity - infinity, and 0^0.
(More precisely, the
limits of these quantities are indeterminate forms; the quantities themselves are always undefined.)
Indeterminate forms can have various meanings depending on how you got the values in them. 0/0 as such is meaningless -- it's
undefined. But in a limit,
X*5/X = 5, _even if X is 0_! That is, X*5/X, when X is 0, becomes 0/0, which is undefined; but the limit of X*5/X as X approaches 0 is 5, because you are allowed to cancel the X's. Thus, 0/0 can have different
values depending on where the 0's came from, so it is called
indeterminate.
The other types of indeterminate forms are harder to deal with, but the same logic applies.