The value of $f(0)$, so that the function $f(x) = \frac{\sqrt{a^{2} - ax + x^{2}} - \sqrt{a^{2} + ax + x^{2}}}{\sqrt{a + x} - \sqrt{a - x}}$ Becomes continuous for all $x$, is given by

Question

Dr. Abubacker · Accepted Answer

(c)
$f(x) = \frac{\sqrt{a^{2} - ax + x^{2}} - \sqrt{a^{2} + ax + x^{2}}}{\sqrt{a + x} - \sqrt{a - x}} 	imes \frac{\sqrt{a^{2} - ax + x^{2}} + \sqrt{a^{2} + ax + x^{2}}}{\sqrt{a^{2} - ax + x^{2}} + \sqrt{a^{2} + ax + x^{2}}} 	imes \frac{\sqrt{a + x} + \sqrt{a - x}}{\sqrt{a + x} + \sqrt{a - x}}$
$\Rightarrow \lim_{x ightarrow 0}\ f(x)$
$= \lim_{x ightarrow 0}\ \frac{- 2ax(\sqrt{a + x} + \sqrt{a - x})}{2x\ (\sqrt{a^{2} - ax + x^{2}} + \sqrt{a^{2} + ax + x^{2})}}$
$= \frac{- a(2\sqrt{a})}{(a + a)} = - \sqrt{a}$

Q. The value of $f(0)$, so that the function $f(x) = \frac{\sqrt{a^{2} - ax + x^{2}} - \sqrt{a^{2} + ax + x^{2}}}{\sqrt{a + x} - \sqrt{a - x}}$ Becomes continuous for all $x$, is given by

Explanation

Reviewed by Dr. Abubacker

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