lim-x-a-x-a-x-a-x-2-a-2-

Question Number 149239 by john_santu last updated on 04/Aug/21

\lim_{x \to a^{+}} \frac{\sqrt{x} - \sqrt{a} - \sqrt{x - a}}{\sqrt{x^{2} - a^{2}}} = ?

Answered by EDWIN88 last updated on 04/Aug/21

\lim_{x \to a^{+}} \frac{{(\sqrt{x - a})}^{2}}{(\sqrt{x} + \sqrt{a}) \sqrt{(x - a) (x + a)}} - \lim_{x \to a^{+}} \frac{\sqrt{x - a}}{\sqrt{x - a} \sqrt{x + a}}

= \lim_{x \to a^{+}} \frac{\sqrt{x - a}}{(\sqrt{x} + \sqrt{a}) \sqrt{x + a}} - \lim_{x \to a^{+}} \frac{1}{\sqrt{x + a}}

= 0 - \frac{1}{\sqrt{2 a}} = - \frac{1}{\sqrt{2 a}}

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