Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

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

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

#### Solution

$\lim_{x \to a} \frac{\cos x - \cos a}{x - a}$
$= \lim_{x \to a} \frac{- 2 \sin \left( \frac{x + a}{2} \right) \sin \left( \frac{x - a}{2} \right)}{2\left( \frac{x - a}{2} \right)} \left[ \because \cos A - \cos B - 2 \sin \left( \frac{A - B}{2} \right) \sin \left( \frac{A + B}{2} \right) \right]$
$= \lim_{x \to a} - \sin \left( \frac{x + a}{2} \right) \left[ \because \lim_\theta \to a \sin\frac{\left( \theta - a \right)}{\left( \theta - a \right)} = 1 \right]$
$= - \sin \left( \frac{2a}{2} \right)$
$\Rightarrow - \sin a$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.8 | Q 5 | Page 62