Calculate: $\lim\limits_{x\to 0}{\frac{ \sqrt{x+\sqrt{x+\sqrt{x}}} }{ 2\sqrt{x}+4\sqrt[8]{x}}}$ Calculate: $$\lim\limits_{x\to 0}{\frac{ \sqrt{x+\sqrt{x+\sqrt{x}}} }{  2\sqrt{x}+4\sqrt[8]{x}}}$$
In the numerator, by factoring out x I was able to get to: $$\sqrt{x}\sqrt{1 + \frac{1}{\sqrt{x}}}\sqrt[4]{1 + \frac{1}{\sqrt{x}}}$$
In the denumerator my success is:
$$\sqrt{x}(2 + 4\sqrt[-\frac{3}{8}]{x})$$
So I could cancel out $$\frac{\sqrt{x}}{\sqrt{x}}$$
But it will leave me with 
$$\lim\limits_{x\to 0}{\frac{\sqrt{1 + \frac{1}{\sqrt{x}}}\sqrt[4]{1 + \frac{1}{\sqrt{x}}}}{2 + 4\sqrt[-\frac{3}{8}]{x}}}$$
What to do next?

Being lazy (to me, this is not a sin !), let $x=y^8$ with $y >0$. Then, after simplifications,
$$\lim\limits_{x\to 0}{\frac{ \sqrt{x+\sqrt{x+\sqrt{x}}} }{ 2\sqrt{x}+4\sqrt[8]{x}}}=\lim\limits_{y\to 0}\frac{\sqrt{y^6+\sqrt{1+y^4}}}{4+2 y^3}\to \frac 14$$
