lim _{x rightarrow 0} frac{cos 2x - cos 3x}{c | vedclass

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(A) Using $L'H\hat{o}pital's$ rule,we differentiate the numerator and denominator with respect to $x$: $\lim _{x \rightarrow 0} \frac{\frac{d}{dx}(\co

Vedclass · Accepted Answer

$\frac{5}{9}$

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(A) Using $L'H\hat{o}pital's$ rule,we differentiate the numerator and denominator with respect to $x$: $\lim _{x \rightarrow 0} \frac{\frac{d}{dx}(\cos 2x - \cos 3x)}{\frac{d}{dx}(\cos 4x - \cos 5x)} = \lim _{x \rightarrow 0} \frac{-2 \sin 2x + 3 \sin 3x}{-4 \sin 4x + 5 \sin 5x}$ Since this is still in the $\frac{0}{0}$ form,we apply $L'H\hat{o}pital's$ rule again: $\lim _{x \rightarrow 0} \frac{-4 \cos 2x + 9 \cos 3x}{-16 \cos 4x + 25 \cos 5x}$ Substituting $x = 0$: $= \frac{-4 \cos(0) + 9 \cos(0)}{-16 \cos(0) + 25 \cos(0)} = \frac{-4(1) + 9(1)}{-16(1) + 25(1)} = \frac{5}{9}$

$\lim _{x \rightarrow 0} \frac{\cos 2x - \cos 3x}{\cos 4x - \cos 5x} = $

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