Yeddi - the repeating nine in our example, though infinite, remains a symptom of division and this has a limit.
Say you want to go down to the corner store and buy a Pepsi. You think to yourself: since the store is 10 miles away, 5 miles is the halfway point between your body and that refreshing filthwater of capitalism. You conclude that you will invariably cross that halfway mark before you reach the store. Seems logical enough on the surface.
So, you start driving down the street.
Sure enough, you pass the 5 mile halfway point on the way to the store. You're 5 miles into the trip, and you think to yourself that the halfway point between you and the store is now 2.5 miles. If your earlier logic holds, you posit, then you must also cross this halfway point before you get there. As you continue to drive down the street (you haven't stopped moving, remember), you eventually travel another 2.5 miles and are now 2.5 miles from the store. You realize, at this point, that the new halfway point between you and the store is 1.25 miles. As you continue traveling, another 1.25 miles goes by. The halfway point between you and the store is now .625 miles. Continuing on, once you've traveled an additional .625 miles, the new halfway point between you and the store is .3125 miles...
By your logic, applied to our {.99...}, you would never get to the store. You'd get really close, but for all of eternity you'd just fall short of it because you've always got to cross that halfway point between you and the store before you get there.
This, as we know from the fact that we can move at all, is false. You'll get there.
9/9 = {.99...} = 1
What you're thinking of is the philosophical concept of an infinite number - which, really, isn't math. Calculus doesn't work that way.