I'm still not convinced that you can't just say that x/0 equals infinity.Certainly, there's nothing to stop you from saying that. We could quite easily define a number called Infinity (or, say, Frank) such that x/0 = Frank, the same way we defined a number i such that sqrt(-1) = i. People have thought about this for centuries, I'm sure, but the reason that it hasn't taken off the way i has is that positing the existence of "i" gets you a whole lot of new and useful mathematics (e.g. a new field of complex numbers which is closed under arbitrary roots) whereas positing the existence of Frank doesn't buy you much except trouble.
For instance, consider our new number system consisting of the reals plus Frank. We'll have to define operations on Frank now. Clearly, Frank + Frank = Frank and Frank * Frank = Frank. But what about Frank - Frank and Frank/Frank? You can't define them as 0 and 1 respectively, or else you find yourself with proofs like this:
Frank = Frank + Frank
Frank/Frank = Frank/Frank + Frank/Frank
1 = 2
which is undesirable. So you've still got undefined numbers in your number system, and your number system no longer has lots of the nice properties that you get when you have a field.
Also, Frank doesn't share all that many of the properties that are usually associated with what we now know about infinities -- e.g. the idea that there are "different sizes" of infinite sets which can't be mapped onto one another.