What is $\lim_{n\to\infty} \frac{x_n}{y_n}$ provided $x_n=2y_{n}-y_{n-1}$ and $y_n=3y_{n-2}-y_{n-2}$ with $x_0=y_0=1$?

Hint: Let $z_{n}\equiv(x_{n},y_{n})^{\intercal}$. Then, we can rewrite the recurrence as $$ z_{n}=Mz_{n-1}\text{ (}n\geq1\text{)} $$ where $$ M\equiv\begin{pmatrix}2 & 1\\ 1 & 1 \end{pmatrix}. $$ Note that $M$ is diagonalizable. That is, we can write $M=SJS^{-1}$ where $$ S\equiv\begin{pmatrix}\frac{1}{2}\left(1-\sqrt{5}\right) & \frac{1}{2}\left(1+\sqrt{5}\right)\\ 1 & 1 \end{pmatrix}\text{ and }J\equiv\begin{pmatrix}\frac{1}{2}\left(3-\sqrt{5}\right)\\ & \frac{1}{2}\left(3+\sqrt{5}\right) \end{pmatrix} $$ By induction, we have $z_{n}=M^{n}z_{0}$. Note that $$ M^{n}=\left(SJS^{-1}\right)^{n}=SJ^{n}S^{-1}. $$ Can you figure out the rest?