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Question
\[\lim_{x \to 0} \frac{ax + x \cos x}{b \sin x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{ax + x \cos x}{b \sin x} \right]\]
\[\text{ Dividing the numerator and the denominator by } x:\]
\[ \lim_{x \to 0} \left[ \frac{a + \cos x}{b\left( \frac{\sin x}{x} \right)} \right] \left[ \because \lim_{x \to 0} \frac{\sin x}{x} = 1 \right]\]
\[ = \frac{a + \cos 0}{b}\]
\[ = \frac{a + 1}{b}\]
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