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Find the limit of the given function

Find the value of the following limit.

    \[ \lim_{x \to a^+} \frac{ \sqrt{x} - \sqrt{a} + \sqrt{x-a}}{\sqrt{x^2-a^2}}. \]


We compute as follows,

    \begin{align*}  \lim_{x \to a^+} \frac{\sqrt{x} - \sqrt{a} + \sqrt{x-a}}{\sqrt{x^2-a^2}} &= \lim_{x \to a^+} \left( \frac{\sqrt{x} - \sqrt{a}}{\sqrt{x^2 - a^2}} + \frac{\sqrt{x-a}}{\sqrt{(x+a)(x-a)}} \right) \\[9pt]  &= \lim_{x \to a^+} \left( \frac{\sqrt{x} - \sqrt{a}}{\sqrt{x^2 - a^2}} \right) + \frac{1}{2\sqrt{a}} \\[9pt]  &= \frac{1}{2 \sqrt{a}} + \lim_{x \to a^+} \frac{ \frac{1}{2 \sqrt{x}}}{\frac{x}{\sqrt{x^2-a^2}}} &(\text{L'Hopital})\\[9pt]  &= \frac{1}{2\sqrt{a}} + \lim_{x \to a^+} \frac{\sqrt{x^2-a^2}}{2x^{\frac{3}{2}}} \\[9pt]  &= \frac{1}{2\sqrt{a}} + 0 = \frac{1}{2\sqrt{a}}. \end{align*}

Since this limit exists, the application of L’Hopital’s rule was justified.

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