Limit of term with roots How can I prove the following:
$$\lim_{k\rightarrow \infty}  \frac{2}{3 (\sqrt{k+1} - \sqrt{k-1})(\sqrt{k+2} + \sqrt{k})} = \frac{1}{3}$$

We multiply by the conjugate of $\sqrt{k+1} - \sqrt{k-1}$ and we use the asymptotic limit
$$\sqrt{k+a}\sim_\infty \sqrt k$$
we find
$$\lim_{k\rightarrow \infty}  \frac{2}{3 (\sqrt{k+1} - \sqrt{k-1})(\sqrt{k+2} + \sqrt{k})} =\lim_{k\rightarrow \infty}\frac23\frac{2\sqrt k}{4\sqrt k}=\frac13$$
