Explanation

(a)
$\left( \lim_{x \rightarrow \infty}\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x} \times \frac{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}}{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}} \right)$
$= \lim_{x \rightarrow \infty}\left( \frac{\sqrt{x + \sqrt{x}}}{\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}} \right)$
$= \lim_{y \rightarrow 0}\frac{(\sqrt{1 + \sqrt{y})}/\sqrt{y}}{\sqrt{\frac{1}{y} + \frac{\sqrt{1 + \sqrt{y}}}{\sqrt{y}}} + \sqrt{\frac{1}{y}}}$ $\left\lbrack put\ \ x = \frac{1}{y} \right\rbrack$
$= \lim_{y \rightarrow 0}\frac{\sqrt{1 + \sqrt{y}}}{\sqrt{1 + \sqrt{y(1 + \sqrt{y}})} + 1} = \frac{1}{2}$