# Lim X → a Cos X − Cos a √ X − √ a - Mathematics

$\lim_{x \to a} \frac{\cos x - \cos a}{\sqrt{x} - \sqrt{a}}$

#### Solution

$\lim_{x \to a} \frac{\cos x - \cos a}{\sqrt{x} - \sqrt{a}}$
$= \lim_{x \to a} \frac{- 2 \sin \left( \frac{x + a}{2} \right) \sin \left( \frac{x - a}{2} \right)}{\sqrt{x} - \sqrt{a}}$
$\text{ Dividing the numerator and the denominator by }\sqrt{x} + \sqrt{a}:$
$- 2 \lim_{x \to a} \frac{\left( \sin \left( \frac{x + a}{2} \right)\sin \left( \frac{x - a}{2} \right) \right)}{2\left( \frac{x - a}{2} \right)}\left( \sqrt{x} + \sqrt{a} \right)$
$= - 2 \lim_{x \to a} \sin \left( \frac{x + a}{2} \right) \times \frac{\sin \left( \frac{x - a}{2} \right)}{2\left( \frac{x - a}{2} \right)}\left( \sqrt{x} + \sqrt{a} \right)$
$\Rightarrow - 2 \sin \left( \frac{2a}{2} \right) \left( \frac{\sqrt{a} + \sqrt{a}}{2} \right)$
$= - 2\sqrt{a} \sin a$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.8 | Q 14 | Page 62