Why does $\lim_{x\to 0+} \sqrt{\frac {1}{x}+2}-\sqrt{\frac {1}{x}}$ equal zero? Limit $$\lim_{x\to 0+} \sqrt{\frac {1}{x}+2}-\sqrt{\frac {1}{x}}$$
equals zero.
Could you help me prove it?

Let $y = \dfrac1x$. We then have
$$L = \lim_{x \to 0^+} \left(\sqrt{\dfrac1x+2} - \sqrt{\dfrac1x}\right) = \lim_{y \to \infty} \left(\sqrt{y+2} - \sqrt{y}\right) = \lim_{y \to \infty} \left(\sqrt{y+2} - \sqrt{y}\right) \times \dfrac{\sqrt{y+2} + \sqrt{y}}{\sqrt{y+2} + \sqrt{y}}$$
This gives us
$$L = \lim_{y \to \infty} \dfrac{y+2 - y}{\sqrt{y+2} + \sqrt{y}} = \lim_{y \to \infty} \dfrac2{\sqrt{y+2} + \sqrt{y}}$$
Can you now finish it off?
