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Question
\[\lim_{x \to 0} \frac{\sin 3x + 7x}{4x + \sin 2x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{\sin 3x + 7x}{4x + \sin 2x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\frac{\sin 3x}{3x} \times 3x + 7x}{4x + \frac{\sin 2x}{2x} \times 2x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\left( \frac{\sin 3x}{3x} \times 3 + 7 \right)x}{\left( 4 + \frac{\sin 2x}{2x} \times 2 \right)x} \right]\]
\[ = \frac{3 + 7}{4 + 2}\]
\[ = \frac{10}{6} = \frac{5}{3}\]
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