1. I think that means domain (all possible x values). For example, the |x|<1 after the function.
2. Think of a binomial expansion with increasing powers of x. What if |x|<1? As an arbitrary variable 'k' tends to infinity, the value of x^k tends to 0. Therefore, the sum of the first 'k' terms of the binomial expansion (1+x)^n approaches (as long as k<n) (1+x)^n.
This is especially apparent if x=10^-k. With each 'new' term of the binomial expansion, you're increasing your accuracy by 'k' decimal points because you're getting 10^k times closer to the exact value with each 'new' term.
As for where they got the 1/10 from, it's given in (a) ^^;