$\displaystyle \lim_{x\rightarrow \infty }\sqrt{x+\sqrt{x}}+\sqrt{x}$
May be that there is a '+' instead of a '-' and the limit is...
$\displaystyle \displaystyle \lim_{x \rightarrow \infty} \sqrt{x + \sqrt{x}} - \sqrt {x}$ (1)
In that case is...
$\displaystyle \displaystyle \lim_{x \rightarrow \infty} \sqrt{x + \sqrt{x}} - \sqrt {x}= \lim_{x \rightarrow \infty} \frac{\sqrt{x}}{\sqrt{x + \sqrt{x}} + \sqrt {x}} = $
$\displaystyle \displaystyle = \lim_{x \rightarrow \infty} \frac{1}{1 + \sqrt{1 + \frac{1}{\sqrt{x}}}} = \frac{1}{2}$ (2)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$