Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\cos{3x}-\cos{4x}}{x\sin{2x}}}$ by L’Hospital’s Rule

$\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{(\cos{3x}-\cos{4x})’}{(x\sin{2x})’}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d}{dx}(\cos{3x}-\cos{4x})}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d}{dx}\cos{3x}-\dfrac{d}{dx}\cos{4x}}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d}{dx} \times 1 \times \cos{3x}-\dfrac{d}{dx} \times 1 \times \cos{4x}}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d}{dx} \times \dfrac{d(3x)}{d(3x)} \times \cos{3x}-\dfrac{d}{dx} \times \dfrac{d(4x)}{d(4x)} \times \cos{4x}}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d \times d(3x)}{dx \times d(3x)} \times \cos{3x}-\dfrac{d \times d(4x)}{dx \times d(4x)} \times \cos{4x}}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d(3x) \times d}{dx \times d(3x)} \times \cos{3x}-\dfrac{d(4x) \times d}{dx \times d(4x)} \times \cos{4x}}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d(3x)}{dx} \times \dfrac{d}{d(3x)} \times \cos{3x}-\dfrac{d(4x)}{dx} \times \dfrac{d}{d(4x)} \times \cos{4x}}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d(3x)}{dx} \times \dfrac{d}{d(3x)}\cos{3x}-\dfrac{d(4x)}{dx} \times \dfrac{d}{d(4x)}\cos{4x}}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d(3x)}{dx} \times (-\sin{3x})-\dfrac{d(4x)}{dx} \times (-\sin{4x})}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d}{dx}(3x) \times (-\sin{3x})-\dfrac{d}{dx}(4x) \times (-\sin{4x})}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\dfrac{d}{dx}(3 \times x) \times (-\sin{3x})-\dfrac{d}{dx}(4 \times x) \times (-\sin{4x})}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{3 \times \dfrac{d}{dx}(x) \times (-\sin{3x})-4 \times \dfrac{d}{dx}(x) \times (-\sin{4x})}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{3 \times \dfrac{dx}{dx} \times (-\sin{3x})-4 \times \dfrac{dx}{dx} \times (-\sin{4x})}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{3 \times 1 \times (-\sin{3x})-4 \times 1 \times (-\sin{4x})}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{3 \times (-\sin{3x})-4 \times (-\sin{4x})}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{-3\sin{3x}+4\sin{4x}}{\dfrac{d}{dx}(x\sin{2x})}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{-3\sin{3x}+4\sin{4x}}{x \times \dfrac{d}{dx}\sin{2x}+\sin{2x} \times \dfrac{d}{dx}x}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{-3\sin{3x}+4\sin{4x}}{x \times \dfrac{d}{dx} \times 1 \times \sin{2x}+\sin{2x} \times \dfrac{dx}{dx}}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{-3\sin{3x}+4\sin{4x}}{x \times \dfrac{d}{dx} \times \dfrac{d(2x)}{d(2x)} \times \sin{2x}+\sin{2x} \times \dfrac{dx}{dx}}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{-3\sin{3x}+4\sin{4x}}{x \times \dfrac{d \times d(2x)}{dx \times d(2x)} \times \sin{2x}+\sin{2x} \times \dfrac{dx}{dx}}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{-3\sin{3x}+4\sin{4x}}{x \times \dfrac{d(2x) \times d}{dx \times d(2x)} \times \sin{2x}+\sin{2x} \times \dfrac{dx}{dx}}}$

$=\,\,$ $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{-3\sin{3x}+4\sin{4x}}{x \times \dfrac{d(2x)}{dx} \times \dfrac{d}{d(2x)} \times \sin{2x}+\sin{2x} \times \dfrac{dx}{dx}}}$

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