limits question with radicals, rationalizing

Question

Find the limit value

Here's what I did (Above)

I think I can rationalize the numerator to solve it, but I'm having trouble rationalizing numerator, when I'm usually rationalizing the denominator.

How do I rationalize the numerator? (If I'm on the right track for solution)

Answer 1

$$\begin{align}\lim_{x\to0}\frac{\sqrt{x+2}-\sqrt2}{x}&= \lim_{x\to0}\frac{\sqrt{x+2}-\sqrt2}{x}\cdot\frac{\sqrt{x+2}+\sqrt2}{\sqrt{x+2}+\sqrt2}\\ &=\lim_{x\to0}\frac{x+2-2}{x(\sqrt{x+2}+\sqrt2)}\\ &=\lim_{x\to0}\frac{x}{x(\sqrt{x+2}+\sqrt2)}\\ &=\lim_{x\to0}\frac{1}{\sqrt{x+2}+\sqrt2}\\ &=\frac{1}{2\sqrt2}\\ \end{align}$$

Answer 2

$$\lim_{x\to0}\frac{\sqrt{x+2}-\sqrt2}{x} = \lim_{x\to0}\frac{\sqrt{2+x}-\sqrt2}{x-0} = (\sqrt{t})'\big|_{t=2}=\frac{1}{2\sqrt{t}}\big|_{t=2}=\frac{\sqrt{2}}{4}.$$