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)

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