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Question
\[\lim_{x \to 0} \frac{5x + 4 \sin 3x}{4 \sin 2x + 7x}\]
Solution
\[ = \lim_{x \to 0} \left[ \frac{5x + 4 \times \frac{\sin 3x}{3x} \times 3x}{\frac{4 \sin 2x}{2x} \times 2x + 7x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\left( 5 + 4 \frac{\sin 3x \times 3}{3x} \right)x}{\left( 4\frac{\sin 2x}{2x} \times 2 + 7 \right)x} \right]\]
\[ = \frac{5 + 4 \times 3}{4 \times 2 + 7} \left[ \because \lim_{x \to 0} \frac{\sin \left( 3x \right)}{3x} = 1 \right]\]
\[ = \frac{17}{15}\]
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