lim-x-a-x-a-x-a-x-2-a-2-a-gt-0-

Question Number 195385 by Rodier97 last updated on 01/Aug/23

\lim_{x \Rightarrow a^{+}} \frac{\sqrt{x} - \sqrt{a} - \sqrt{x - a}}{\sqrt{x^{2} - a^{2}}}; a > 0

Answered by cortano12 last updated on 01/Aug/23

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

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

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