Question
$\mathop {\lim }\limits_{x \to \infty } \sqrt {{a^2}{x^2} + ax + 1} - \sqrt {{a^2}{x^2} + 1} =$
$ = \mathop {\lim }\limits_{x \to \infty } \,\frac{{ax}}{{\,\sqrt {{a^2}{x^2} + ax + 1} + \sqrt {{a^2}{x^2} + 1} }}$
$ = \mathop {\lim }\limits_{x \to \infty } \,\frac{a}{{\,\sqrt {{a^2} + \frac{a}{x} + \frac{1}{{{x^2}}}} + \sqrt {{a^2} + \frac{1}{{{x^2}}}} }} = \frac{a}{{2a}} = \frac{1}{2}$.
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