The value of limx→0 a2−ax+x2−a2+ax+x2a+x−a−x ,is

A
$a$
B
$\sqrt{a}$
C
$- a$
D
$- \sqrt{a}$

Solution:

We have,

$\begin{array}{l} lim_{x \to 0} \frac{\sqrt{a^{2} - a x + x^{2}} - \sqrt{a^{2} + a x + x^{2}}}{\sqrt{a + x} - \sqrt{a - x}} \\ = lim_{x \to 0} \frac{- 2 a x (\sqrt{a + x} + \sqrt{a - x})}{2 x (\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}})} \\ = lim_{x \to 0} \frac{- a (\sqrt{a + x} + \sqrt{a - x})}{\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}}} = - \frac{2 a^{3 / 2}}{2 a} = - \sqrt{a} \end{array}$

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