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

$$$$$$$$$${\lim }_{x\Rightarrow {\mathrm{a}}^{+}}\frac{\sqrt{x}-\sqrt{\mathrm{a}}-\sqrt{x-\mathrm{a}}}{\sqrt{x^2-{\mathrm{a}}^2}};\mathrm{a}>0$$

Answered by cortano12 last updated on 01/Aug/23

$$=\underset{x\to {\mathrm{a}}^{+}}{\lim }\left(\frac{\frac{\mathrm{x}-\mathrm{a}}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{a}}}-\sqrt{\mathrm{x}-\mathrm{a}}}{\sqrt{\mathrm{x}+\mathrm{a}}\sqrt{\mathrm{x}-\mathrm{a}}}\right)$$$$=\underset{x\to {\mathrm{a}}^{+}}{\lim }\left(\frac{\frac{\sqrt{\mathrm{x}-\mathrm{a}}}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{a}}}-1}{\sqrt{\mathrm{x}+\mathrm{a}}}\right)$$$$=\frac{0-1}{\sqrt{2\mathrm{a}}}=-\frac{1}{\sqrt{2\mathrm{a}}}$$

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