# Lim X → 0 Cos 3 X − Cos 5 X X 2 - Mathematics

Sum
$\lim_{x \to 0} \frac{\cos 3x - \cos 5x}{x^2}$

#### Solution

$\lim_{x \to 0} \left[ \frac{\cos 3x - \cos 5x}{x^2} \right]$
$= \lim_{x \to 0} \left[ \frac{- 2\sin\left( \frac{3x + 5x}{2} \right) \sin\left( \frac{3x - 5x}{2} \right)}{x^2} \right] \left[ \because cosC - cosD = - 2\sin\left( \frac{C + D}{2} \right)\sin\left( \frac{C - D}{2} \right) \right]$
$= 2 \lim_{x \to 0} \left[ \frac{2\sin 4x \sin x}{x^2} \right] \left[ \because \sin\left( - \theta \right) = - \sin\theta \right]$
$= 2 \lim_{x \to 0} \left[ \frac{\sin 4x}{4x} \times 4 \times \frac{\sin x}{x} \right]$
$= 2 \times 4$
$= 8 .$
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.7 | Q 24 | Page 50