Explanation

(a)
$\lim_{x \rightarrow \infty}{\sqrt{a^{2}x^{2} + ax + 1} - \sqrt{a^{2}x^{2} + 1}}$
$= \lim_{x \rightarrow \infty}\frac{a^{2}x^{2} + ax + 1 - a^{2}x^{2} - 1}{\sqrt{a^{2}x^{2} + ax + 1} + \sqrt{a^{2}x^{2} + 1}}$
$= \lim_{x \rightarrow \infty}\frac{ax}{x\left\lbrack \sqrt{a^{2} + \frac{a}{x} + \frac{1}{x^{2}}} + \sqrt{a^{2} + \frac{1}{x^{2}}} \right\rbrack}$
$= \frac{a}{\sqrt{a^{2}} + \sqrt{a^{2}}} = \frac{1}{2}$