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Question
\[\lim_{x \to 0} \frac{\cos 3x - \cos 7x}{x^2}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{\cos 3x - \cos 7x}{x^2} \right]\]
\[= \lim_{x \to 0} \left[ \frac{- 2\sin\left( \frac{3x + 7x}{2} \right)\sin\frac{\left( 3x - 7x \right)}{2}}{x^2} \right] \left[ \cos C - \cos D = - 2\sin\left( \frac{C + D}{2} \right)\sin\left( \frac{C - D}{2} \right) \right]\]
\[ = \lim_{x \to 0} \left[ \frac{- 2\sin 5x \sin \left( - 2x \right)}{x^2} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2\sin 5x \sin 2x}{x^2} \right] \left[ \because \sin\left( - \theta \right) = - \sin\theta \right]\]
\[ = 2 \lim_{x \to 0} \left[ \frac{\sin 5x}{5x} \times \frac{\sin 2x}{2x} \right] \times 5 \times 2\]
\[\]
\[ = 2 \times 5 \times 2\]
\[ = 20\]
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