The value of mathop {lim }limits_{x to 0} lef | vedclass

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Question

(D) To evaluate the limit $\mathop {\lim }\limits_{x 	o 0} \left[ \frac{\sqrt{a + x} - \sqrt{a - x}}{x} ight]$,we rationalize the numerator:$\matho

Vedclass · Accepted Answer

$1/\sqrt{a}$

Vedclass · Answer

(D) To evaluate the limit $\mathop {\lim }\limits_{x 	o 0} \left[ \frac{\sqrt{a + x} - \sqrt{a - x}}{x} ight]$,we rationalize the numerator:$\mathop {\lim }\limits_{x 	o 0} \left[ \frac{(\sqrt{a + x} - \sqrt{a - x})(\sqrt{a + x} + \sqrt{a - x})}{x(\sqrt{a + x} + \sqrt{a - x})} ight]$$= \mathop {\lim }\limits_{x 	o 0} \left[ \frac{(a + x) - (a - x)}{x(\sqrt{a + x} + \sqrt{a - x})} ight]$$= \mathop {\lim }\limits_{x 	o 0} \left[ \frac{2x}{x(\sqrt{a + x} + \sqrt{a - x})} ight]$$= \mathop {\lim }\limits_{x 	o 0} \left[ \frac{2}{\sqrt{a + x} + \sqrt{a - x}} ight]$$= \frac{2}{\sqrt{a + 0} + \sqrt{a - 0}} = \frac{2}{2\sqrt{a}} = \frac{1}{\sqrt{a}}$.

The value of $\mathop {\lim }\limits_{x \to 0} \left[ \frac{\sqrt{a + x} - \sqrt{a - x}}{x} \right]$ is

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