Asymptotic ratio with two variables

Question

What's the thought process when you try to get the asymptotic ratio with two variables x and y?

$\lim_{n \to \infty} \frac{x^{n+1} - y^{n+1}}{2x^{n} - 2y^{n}}$

Answer 1

Suppose $x$ and $y$ are positive. If $x>y$, then, for large $n$, $x^n$ will be much larger than $y^n$. As a result, the numerator will be "about" $x^{n+1}$ and the denominator will be "about" $2x^n$, and so the ratio will be "about" $x/2$. As $n$ tends to infinity, this is indeed the limit.

You can work this out by dividing by $x^n$ to write the ratio as $$ \frac{x-y\left( \frac{y}{x} \right)^n }{2-2\left( \frac{y}{x} \right)^n } $$ and then considering what happens to $(y/x)^n$ as $n$ goes to infinity.

Consider the $y>x$ case similarly.