11. Given
$$
(1+\sqrt{3})^{n}=a_{n}+b_{n} \sqrt{3} \text {, }
$$

where $a_{n}$ and $b_{n}$ are integers. Then $\lim _{n \rightarrow+\infty} \frac{a_{n}}{b_{n}}=$ $\qquad$ .
11. $\sqrt{3}$.

From the given, we know
$$
(1-\sqrt{3})^{n}=a_{n}-b_{n} \sqrt{3} \text {. }
$$

Then $a_{n}=\frac{1}{2}\left[(1+\sqrt{3})^{n}+(1-\sqrt{3})^{n}\right]$,
$$
b_{n}=\frac{1}{2 \sqrt{3}}\left[(1+\sqrt{3})^{n}-(1-\sqrt{3})^{n}\right] \text {. }
$$

Therefore, $\lim _{n \rightarrow+\infty} \frac{a_{n}}{b_{n}}$
$$
=\lim _{n \rightarrow+\infty} \sqrt{3} \times \frac{(1+\sqrt{3})^{n}+(1-\sqrt{3})^{n}}{(1+\sqrt{3})^{n}-(1-\sqrt{3})^{n}}=\sqrt{3}
$$