Question

lim sin2x-tan3x/x
x-> o

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Answer 1

$\lim_{x\to 0} \frac{\sin2x- \tan3x}{x}= \lim_{x\to 0} \frac{\sin2x}{x}-\lim_{x\to 0} \frac{\tan3x}{x}\\\\=\lim_{x\to 0}\frac{2\sin2x}{2x}-\lim_{x\to 0}\frac{\sin3x}{x\cos3x}\\\\=2\lim_{x\to 0}\frac{\sin2x}{2x}-\lim_{x\to 0}\frac{\sin3x}{x}\times \lim_{x \to 0} \frac{1}{\cos3x} \\\\=2\times 1-\lim_{x\to 0}\frac{3\sin3x}{3x}\times \lim_{x \to 0} \frac{1}{\cos3x}\\=2-3\lim_{x\to 0}\frac{\sin3x}{3x}\times \frac{1}{\cos0}\\=2-3\times 1\times 1\\=2-3\\=-1$