1. Given $(1+\sqrt{3})^{n}=a_{n}+b_{n} \sqrt{3}$, where $a_{n}$ and $b_{n}$ are integers. Then $\lim _{n \rightarrow+\infty} \frac{a_{n}}{b_{n}}=$ $\qquad$
$(2011$, National High School Mathematics League Sichuan Preliminary Competition)
Hint: $\lim _{n \rightarrow+\infty} \frac{a_{n}}{b_{n}}$ $=\lim _{n \rightarrow+\infty} \sqrt{3} \times \frac{(1+\sqrt{3})^{n}+(1-\sqrt{3})^{n}}{(1+\sqrt{3})^{n}-(1-\sqrt{3})^{n}}=\sqrt{3}$.