If I remember correctly

where x != 0

x/0 is undefined

0/0 is worse than undefined, it tells you nothing.

If you remember back to working on rational functions (polynomial over polynomial), if you got 0/0 you knew nothing about the point. However, if you get x/0, you know there's an asymptote.

For example:
Code:
x^2 +  x - 6
------------
x^2 + 2x - 3
Try the following numbers:
{-3,2,1}

-3 : yields 0/0, which tells you nothing about what exists at x = -3
2 : yields 0/-5, or a valid 0
1 : yields -4/0, or undefined -- this tells you that there is a verticle asymptote here

0/0 is more than undefined, it's more like nothingness