Let $a_n$ and $b_n$ be two sequences defined by the recurrence relation \begin{align} a_{n + 1} = \frac{{1 + a_n + a_n b_n }}{{b_n }},\qquad b_{n + 1} = \frac{{1 + b_n + a_n b_n }}{{a_n }} \end{align} with $a_1=1$ and $b_1=2$. Find $\mathop {\lim }\limits_{n \to \infty } a_n$.
Simple manipulations yield that \begin{align} a_n = \frac{{1 + b_n }}{{b_{n + 1} - b_n }},\qquad b_n = \frac{{1 + a_n }}{{a_{n + 1} - a_n }} \end{align} I tried to find the first few terms of $a_n$ and $b_n$ but it seem that they doesn't have a general form. Once can see that both sequences increasing fast and randomly. Also, I tried to insert $b_n$ in $a_n$, but I got a complicated result and it doesn't help or indicate to anything.
Any help is appreciated. Thanks