Determine the limit $\lim_{n \to \infty} \frac{a_n}{b_n}$ where $a_n = 3a_{n-1} + 4a_{n-2}$ with $a_0=0, a_1=1$ and $b_n$ is the sequence defined by $b_n = 4b_{n-1} + 3b_{n-2}$ with $b_0=0, b_1=1$.