Find the value of : $\lim_{ x \to \infty} \left(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$ I need to calculate the limit of the function below:
$$\lim_{ x \to \infty}  \left(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$$
I tried multiplying by the conjugate, substituting $x=\frac{1}{t^4}$, and both led to nothing.

We have by Taylor series
$$ \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}=\frac{\sqrt{x+\sqrt x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}\sim_\infty\frac{\sqrt x}{2\sqrt x}\xrightarrow{x\to\infty}\frac12$$
