Show the sequence converges and find its limit Show the sequence $(\frac{x^n - y^n}{x^n + y^n})$ where $x$, $y \in \mathbb{R}$, $n \in \mathbb{N}$, and $|x| \neq |y|$ converges and find the limit.
I am lost on how to show this sequence converges. Should I consider cases with $x$ and $y$ values and within each case show the sequence is bounded and either increasing or decreasing to use the Monotone convergence theorem? 
 A: $$\frac{x^n-y^n}{x^n+y^n} =\frac{x^n}{x^n+y^n}-\frac{y^n}{x^n+y^n}$$
$$\stackrel{(*)}{=}\frac{1}{1+(y/x)^n}-\frac{1}{1+(x/y)^n}=a_n-b_n.$$
There are two cases(*). On one of then, $a_n \to 1$ and $b_n \to 0$. On the other, $a_n \to 0$ and $b_n \to 1$.
(*) There are also trivial cases $x=0$ or $y=0$, for which the answer is clear.
A: assume  $y\neq 0$.
Your sequence is
$$1-\frac{2}{1+(\frac{x}{y})^n)}$$
so
if $|\frac{x}{y}|<1$ , $(\frac{x}{y})^n$ goes to $0$ and the sequence converges to $1-2=-1$.
else,  $(\frac{x}{y})^n$ goes to $+\infty$  and the limit is $1$.
if$y=0$, it is $1$.
A: Assume that $|x|>|y|\ne 0.$ Then
$$\dfrac{x^n-y^n}{x^n+y^n}=\dfrac{1-\left(\dfrac{y}{x}\right)^n}{1+\left(\dfrac{y}{x}\right)^n}\to 1,$$ as $n\to\infty.$
Assume that $|x|<|y|\ne 0.$ Then
$$\dfrac{x^n-y^n}{x^n+y^n}=\dfrac{\left(\dfrac{x}{y}\right)^n-1}{\left(\dfrac{x}{y}\right)^n+1}\to -1,$$ as $n\to\infty.$
We have used that $a^n\to 0$ as $n\to \infty$ for any $a$ with $|a|<1.$
If $x=0$ or $y=0$ the limit should be clear.
