First slide

Introduction to limits

Question

The value of $lim_{x \to 0} \frac{\sqrt{a^{2} - a x + x^{2}} - \sqrt{a^{2} + a x + x^{2}}}{\sqrt{a + x} - \sqrt{a - x}}$ ,is

Moderate

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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