TITLE: is $0.\overline{99}$ the same as $\lim_{x \to 1} x$?
QUESTION [6 upvotes]: So we had an interesting discussion the other day about 0.999... repeated to infinity, actually being equal to one.  I understand the proof, but I'm wondering then if you had the function...
$$
f(x) = x* \frac{(x-1)}{(x-1)}
$$
so
$$
f(1) = NaN
$$
and 
$$
\lim_{x \to 1} f(x) = 1
$$
what would the following be equal to? 
$$
f(0.\overline{999}) = ?
$$

REPLY [5 votes]: While it's true that $0.\overline{9}=\lim_{x\to 1} x=1$, in your case you may not move the limit inside $f$ to get $$\lim_{x\to 1}f(x)=f(1)$$ since the RHS is undefined. In general, you may move the a limit from outside the function to inside the function only if the function in question is continuous at the point where you are taking the limit.