Why is $\lim\limits_{N\to\infty}x^{N+1}=0$, where $|x|<1$? How is this done? 
Why is $\lim\limits_{N\to\infty}x^{N+1}=0$, where $-1<x<1$?

Given $x \neq 0$ (case equal to zero is trivial) you want $|x^n|<\epsilon \ \ \forall N>n$ for some $N$ you have to find that $N$.
$$|x^n|<\epsilon \iff|x|^n<\epsilon  \iff n \log|x|<\log\epsilon \iff n>\frac{\log \epsilon}{\log|x|} $$
