# How to evaluate $\lim_{x\to 0}\frac{x}{(\sqrt4+\sqrt x)−(\sqrt4-\sqrt x)}$?

I’m absolutely lost on how to do this question.

How to evaluate $$\lim_{x\to 0}\frac{x}{(\sqrt4+\sqrt x)−(\sqrt4-\sqrt x)}?$$

I know that I have to multiply the numerator and the denominator by the conjugate, which should be $${(\sqrt4+\sqrt x) + (\sqrt4-\sqrt x)}$$ But the square roots are throwing me off. I’ve tried separating them individually but then I get a $$\sqrt{-x}$$ for $${(\sqrt4-\sqrt x)}$$ which isn’t possible. I tried putting the minus sign outside of the square root sign but I’m not sure if this is the right way to go about it.

• Since the function is only defined for $\;\sqrt4\neq x>0\;$ , you actually want the limit $\;x\to0^+\;$ (from the right) – DonAntonio Nov 8 '18 at 22:10

If you notice that $$\;(\sqrt4+\sqrt x)-(\sqrt4-\sqrt x)=2\sqrt x\;$$, things are pretty simpler...So simple that it looks almost trivial. Check you copied the question correctly
$$\frac{x}{(\sqrt4+\sqrt x)−(\sqrt4-\sqrt x)}=\frac{x}{2\sqrt x}=\frac{x}{2\sqrt x}\frac{\sqrt x}{\sqrt x}=\ldots$$