In example 2.2 part 2, the author makes two restrictions because according to him we are making n approach infinity twice. The first with the numerator, and the second with the denominator. My first question is will this ALWAYS be the case whenever we have more than one sequence term in an equation? That is placing restrictions for multiple Epsilons per each sequence term?
My second question concerns his choice of restrictions. Letting Epsilon be greater than 0 for the numerator is just standard procedure, but why in the world is he making Epsilon = 1 for the denominator?
Thanks for any info, all!
Concerning your first question, in general, yes, if our expression has several $x_n$'s, then we should deal several times with the fact that $n$ is going to infinity.
Concerning the other question, any $\varepsilon\in(0,2)$ would do and $\varepsilon=1$ is just the more natural choice. But if we take, say, $\varepsilon=\frac23$, that will work too. In that case, we will have that$$\lvert x_n\rvert=\bigl\lvert2+(x_n-2)\bigr\rvert\geqslant2-\frac23=\frac43.$$