I have two sequences $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ for which I know that both sequences are diverging and strictly increasing, $\lim_{n\to\infty}\dfrac{a_n}{a_{n+1}}=\lim_{n\to\infty}\dfrac{b_n}{b_{n+1}}=q<1$ (limits exist) and want to derive the limit $p=\lim_{n\to\infty}\dfrac{b_n}{a_{n}}$, which does also exist. It looks to me like a direct application of the Stolz–Cesàro theorem, but when trying to derive the limit, I arrive at statements like $1=1$, for example by doing for sufficiently large $n$, s.t. $a_n > 0$ and $b_n>0$,
$\dfrac{b_{n+1}-b_{n}}{a_{n+1}-a_{n}} = \dfrac{b_{n+1}}{a_{n+1}-a_{n}}-\dfrac{b_{n}}{a_{n+1}-a_{n}}=\dfrac{a_{n+1}^{-1} b_{n+1}}{1-a_{n}a_{n+1}^{-1}}-\dfrac{b_{n}b_{n+1}^{-1}b_{n+1}a_{n+1}^{-1}}{1-a_{n}a_{n+1}^{-1}}$ which tends to $\dfrac{p}{1-q}-\dfrac{qp}{1-q} = p$ as $n\to\infty$
Do any other ideas come to mind to tackle this problem? $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ both arise from recurrence formulas $a_{n+1} = c_n a_n - d_n a_{n-1}$ and $b_{n+1} = c_n b_n - d_n b_{n-1} + e_n$, where the coefficients are affine in $n$, i.e., for example $c_n = c_1 n + c_2$.