Eigenvalues · Superchallenge problem

Let $A = \begin{pmatrix}1 & 2\\3 & 2\end{pmatrix}$ and write $A^n = \begin{pmatrix}p_n & q_n\\r_n & s_n\end{pmatrix}$. Find $\displaystyle\lim_{n\to\infty}\frac{p_n}{r_n}$.