just take the limit of x/x as x-> 0, it's one. so 0:0 is the same as 1:1
The problem is you could also take the limit of x^2/x or x/x^2 as x->0, and get 0/0 is 0 (for the first one) or undefined (for the second). You could get any number: just take the limit of c*x/x as x->0 to get c.
Notice that the limit of x/x^2 as x->0 is NOT infinity. The limit from the right (that is, restricted to x>0) is infinity, while the limit from the left (x<0) is -infinity. What we really mean by ``the limit is infinity'' is that the function grows without bound. Start treating infinity like a number and things get much more complicated (the rules of algebra start needing exceptions, for example).
IAAM and I'm coming off a semester of explaining calc one to students (and preparing to do it again in the spring term as well).