i would say 1/10^(n+1)

But 1/10^(n+1) isn't a single number. It becomes a single number when you substitute in a number for n, say 1000. But 1/10^(1002) < 10^(1001), so 1/10^1001 can't be the desired single number because it must be smaller than 1/10^k for ALL k, including k = 1002. If you put in 100000000 for n, same thing will happen; 1/10^1000000001 cannot be the desired number. No matter what you put in, it won't work. There is no number greater than 0 that is smaller than all of the infinitely many numbers {1/10, 1/100, 1/1000, ...}. Unless you want to say that .0[bar]1 is a positive number different from 0, but this creates its own problems as I discuss below.

when we think of the function in these terms, as the .9 function approaches .999[bar] or "1" as it appears, so too does this .1 function approach .000[bar]1

.999[bar] does not approach a number just as the .1 function does not approach a number

These two statements seem to contradict each other. Is .000[bar]1 a number or not? Does the .1 function approach it or not?

If .000[bar]1 is a number which is not equal to 0, then its reciprocal 1/.0[bar]1 should be a number too. However its reciprocal is bigger than any number. It's infinite. Do you want to posit that infinity is an ordinary number? If so, let's give it a name, call it H. What is H + H? is it the same as H or is it more? If H + H = H then H = 0, a contradiction, so we must have that H + H is not equal to H. Similarly 3*H is different from H and H+H, and the result is there's all these infinite numbers floating around. That's kind of cool but it also makes arithmetic very complicated (for example, what's H^3/(H^2 + 1)? how can we assign it a value that creates no contradictions?)

Wouldn't it be easier if our number system had no infinite numbers in it? If you'd prefer such a system, the only consistent choice is to make .000[bar]1 = 0.

The number .0[bar]1 can be whatever you want it to be in math, so long as you're consistent. If you say it's 0, you get a number system with no infinite numbers. If you say it's not 0, you get a number system with infinite numbers. A number system with no infinite numbers is easier for most mathematicians to understand, so they choose to say that .0[bar]1 has the value 0. There is another system that has infinite numbers, called the hyperreal numbers, but the introduction of infinite numbers makes it very complex and hard to work with. So we say .0[bar]1 = 0 because it's possible to do so consistently, and it's a lot easier than the alternative.

1 does not appear anywhere in the .9 function on the graph shown below, and niether will 0 appear in the .1 function as a y value.

That's why we say "approaches". You can approach a city by walking closer and closer to it, without ever reaching it. Your distance to the city is a function of time that approaches 0 but never reaches it.