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 $-1<x<1$?
 A: Here is a trick:
Put $|x| = \frac{1}{1 + y} < 1$. We want to show that $x^n \to 0 $. In other words, by definition, for a given $\epsilon > 0$, we want to find an $N$ such that for all $n \geq N$, then $|x^n - 0 | < \epsilon $. To find $N$, notice by Bernoulli's inequality
$$ |x^n| = \frac{1}{(1+y)^n} \leq \frac{1}{1+yn} <\epsilon \iff n \geq \frac{1 - \epsilon}{\epsilon y}$$
So, choosing $N =  $ integer part of $\frac{1 - \epsilon}{y \epsilon} $ gives desired result!
A: Consider thinking about it intuitively.
We know that $-1<x<1$. Let's take a arbitrary $x$ as $x=\frac{1}{2}$. 
Now $(\frac{1}{2})^1=\frac{1}{2}=0.500$,
$(\frac{1}{2})^2=\frac{1}{4}=0.250$
$(\frac{1}{2})^3=\frac{1}{8}=0.125$
$\vdots$
As $n$ gets larger and larger, our value will get smaller and smaller. This is because $-1<x<1$, so raising it to $n^{th}$ power as $n$ gets larger and larger actually makes our value smaller and smaller.
A: If $|x|\lt1$, choose an integer $k\ge\frac{|x|}{1-|x|}$. Then
$$
|x|\le\frac{k}{k+1}
$$
Bernoulli's Inequality says that $\left(1+\frac1k\right)^n\ge1+\frac nk$. Therefore,
$$
|x|^n\le\left(\frac{k}{k+1}\right)^n\le\frac{k}{k+n}
$$
If we wish to make $|x|^n$ smaller than any given $\epsilon\gt0$, choose $n\ge k/\epsilon$. Then,
$$
|x|^n\le\frac{k}{k+n}\le\frac{k}{k+k/\epsilon}=\epsilon\frac{k}{k\epsilon+k}\le\epsilon
$$
A: A relatively "low-tech" way to see the limit must be zero (assuming the limit exists) is to call the limit $L$ and note that
$$
L = \lim_{N \to \infty} x^{N+1}
  = \lim_{N \to \infty} (x \cdot x^{N})
  = x \lim_{N \to \infty} x^{N}
  = xL.
$$
Subtracting and factoring, $(1 - x)L = 0$. Since $x \neq 1$ by hypothesis, it must be that $L = 0$.

To prove the limit exists without descending into $\varepsilon$-land (i.e., using convergence criteria that appear early in an elementary analysis course), note that if $0 \leq x < 1$, then the sequence $(x^{N})_{N=0}^{\infty}$ is non-increasing and bounded below by $0$, so it has a real limit $L$.
To handle the case $-1 < x < 0$, note that $-|x|^{N} \leq x^{N} \leq |x|^{N}$ for all $N \geq 0$ and apply the squeeze theorem.
A: 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 $N$.
$$|x^n|<\epsilon \iff|x|^n<\epsilon  \iff n \log|x|<\log\epsilon \iff n>\frac{\log \epsilon}{\log|x|} $$
