Modern calculus, physics, and engineering disregards x's existence at this point, and x becomes zero, so 0.9repeat = 1. This is the realm I was trained in, so I too, agree that x becomes 0 and so 0.9repeat = 1. Number theory also supports this conclusion as well.
im an EE student, and i also agree with you...... but nowhere in the original proof or answer does it say that they are disregarding the tiny difference between 1 and .9repeating.