Evaluate the algebraic limit $\lim_{n\to\infty} \frac{2n}{\sqrt{n+2}(\sqrt{n+1} + \sqrt{n-1})} $ 
$$\lim_{n\to\infty} \frac{2n}{\sqrt{n+2}(\sqrt{n+1} + \sqrt{n-1})} $$

I know that its value is somewhere in $(0, \infty)$ but I have no idea how to find the exact value.
Update: I've had a mistake in formulation of the limit in the original question so I updated the limit.

$$\frac{2n}{\sqrt{n+1} + \sqrt{n-1}}$$
$$=\frac{1}{\sqrt{n}}\frac{1}{\sqrt{1+\frac{1}{n}} + \sqrt{1-\frac{1}{n}}}$$
$$=2\sqrt{n} \cdot \frac{1}{1+\frac{1}{2n}-\frac{1}{8n^2} + 1-\frac{1}{2n}-\frac{1}{8n^2} + o\left(\frac{1}{n^2}\right)}$$
$$=2\sqrt{n} \cdot \frac{1}{2-\frac{1}{4n^2}+o\left(\frac{1}{n^2}\right)}$$
$$=\sqrt{n} \cdot \left(1+\frac{1}{8n^2}+o\left(\frac{1}{n^2}\right)\right)\underset{n\to +\infty}{\longrightarrow}\infty$$
