I hope this helps. Be sure to write back if I didn't address your concerns, or if you just have other questions. After a while, you'll see a pattern. That's how repeating decimals were discovered. What you really want to know, I think, is how limits work, which is a deep question.
Limits are probably the closest thing to magic in mathematics.
Even now, they retain a faint air of paradox. I think you would probably have a lot of fun working through a good textbook on real analysis.