[tex]l = \lim\limits_{x\to a}\dfrac{\sqrt x - \sqrt a+\sqrt{x-a}}{\sqrt{x^2-a^2}} \\ \\ l = \lim\limits_{x\to a} \dfrac{\sqrt x - \sqrt a}{\sqrt{x^2-a^2}}+\lim\limits_{x\to a}\dfrac{\sqrt{x-a}}{\sqrt{x^2-a^2}}=l_1+l_2\\ \\ \\l_1 = \lim\limits_{x\to a}\dfrac{\sqrt x -\sqrt a}{\sqrt{x^2-a^2}} \overset{\frac{0}{0}}{=} \lim\limits_{x\to a} \dfrac{\dfrac{1}{2\sqrt x}}{\dfrac{2x}{2\sqrt{x^2-a^2}}} =\\ \\ =\lim\limits_{x\to a} \dfrac{\sqrt{x^2-a^2}}{2x\sqrt x} = \dfrac{\sqrt{a^2-a^2}}{2a\sqrt a} = 0[/tex]
[tex]l_2 = \lim\limits_{x\to a} \dfrac{\sqrt{x-a}}{\sqrt{x-a}\cdot \sqrt{x+a}} = \lim\limits_{x\to a} \dfrac{1}{\sqrt{x+a}} = \dfrac{1}{\sqrt{2a}} \\ \\\\ \Rightarrow l = 0+\dfrac{1}{\sqrt{2a}}\\\\ \\\Rightarrow \boxed{l = \dfrac{\sqrt{2a}}{2a}}[/tex]
