Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 105: 16

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Given $$\lim_{x\to 0}\frac{2-\cos 3x-\cos 4x}{x}$$ Then \begin{align*} \lim_{x\to 0}\frac{2-\cos 3x-\cos 4x}{x}&\lim_{x\to 0}\frac{[1-\cos 3x]+[1-\cos 4x]}{x} \\ &=\lim_{x\to 0}\frac{1-\cos 3x}{x}+\lim_{x\to 0}\frac{1-\cos 4x}{x}\\ &=3\lim_{3x\to 0}\frac{1-\cos 3x}{3x}+4\lim_{4x\to 0}\frac{1-\cos 4x}{4x}\\ &=0+0\\ &=0 \end{align*}