Classical problem. By the Binomial Theorem, the numerator is \[\frac{(1+\sqrt{3})^{2n}+(1-\sqrt{3})^{2n}}{2}\]while the denominator is \[\frac{1}{\sqrt{3}} \cdot \frac{(1+\sqrt{3})^{2n}-(1-\sqrt{3})^{2n}}{2}\]so that the limit is $\sqrt{3}$.

Same as above, posting for storage. We require, from the Binomial Theorem, $$\sqrt{3} \lim_{n \to \infty} \cdot \frac{(1+\sqrt{3})^n + (1-\sqrt{3} )^n}{(1+\sqrt{3} )^n - (1-\sqrt{3} )^n} = \boxed{\sqrt{3}}$$ Remark: This is highly motivated by, say, Roots of Unity Filter.