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 limit
s 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 value
s 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.