Convergence and the limit of the sequence $x_n=\frac {a_n}{b_n}$ where $(1+\sqrt3)^n=a_n+b_n\sqrt3, n \ge 1$ with $a_n,b_n$ integers
If $(1+\sqrt{2})^n=a_{n}+b_{n}\sqrt{2}\;,(\forall n\in \mathbb{N}).$Then $\lim_{n\rightarrow \infty}\frac{a_{n}}{b_{n}} = $