limits question with radicals, rationalizing 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)

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