<s>[INST] Help finding the limit of the following function as x tends to 0 $$
\lim_{x\to0}\left(\sqrt{\frac{1}{x} + \sqrt{\frac{1}{x} + \sqrt{\frac{1}{x}}}} - \sqrt{\frac{1}{x} - \sqrt{\frac{1}{x} + \sqrt{\frac{1}{x}}}}\,\right)
$$
From Demidovich 5000 problems in mathematical analysis
 [/INST] You can try with $x=t^2$ (the domain is $x>0$, we assume also $t>0$), so your function becomes
$$
\sqrt{\frac{1}{t^2} + \sqrt{\frac{1}{t^2} + \sqrt{\frac{1}{t^2}}}} - \sqrt{\frac{1}{t^2} - \sqrt{\frac{1}{t^2} + \sqrt{\frac{1}{t^2}}}} =
\frac{\sqrt{1+\sqrt{t^2+t^3}}-\sqrt{1-\sqrt{t^2+t^3}}}{t}
$$
and your limit is reduced to computing a derivative.
You can simplify the task observing that the function is also writable as
$$
\frac{2\sqrt{t^2+t^3}}{t}\frac{1}{\sqrt{1+\sqrt{t^2+t^3}}+\sqrt{1-\sqrt{t^2+t^3}}}
$$
and the limit is essentially immediate.
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