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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.

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

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  • $\begingroup$ Opssss...I didn't notice that!!! :) $\endgroup$ – user Nov 8 '18 at 22:33
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    $\begingroup$ It does seem much simpler, thank you! It was actually the first thing I did, however when I reread the question I was given, I was told to specifically use conjugation. But I think I understand what I did wrong and what to do now - I presumed I needed to use conjugation before simplifying the function and thus, it made my life a little bit harder. This was such a huge help and it’s very much appreciated! $\endgroup$ – Renato Sinclair Nov 8 '18 at 22:34
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HINT

$$\frac{x}{(\sqrt4+\sqrt x)−(\sqrt4-\sqrt x)}=\frac{x}{2\sqrt x}=\frac{x}{2\sqrt x}\frac{\sqrt x}{\sqrt x}=\ldots$$

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