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.

I think you're almost right: if I set $\sqrt{x} = t^2$ I got 
$$
\lim_{t \to \infty} \frac{\sqrt{1+\frac{1}{t^2}}}{\sqrt{1+\frac{1}{t^2}\sqrt{1+\frac{1}{t^2}}} + 1} = \frac{1}{2}
$$
