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Question
\[\lim_{x \to 0} \frac{\sin ax + bx}{ax + \sin bx}\]
Solution
\[\lim_{x \to 0} \left[ \frac{\sin \left( ax \right) + bx}{ax + \sin \left( bx \right)} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\frac{\sin ax}{ax} \times ax + bx}{ax + \frac{\sin bx}{bx} \times bx} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\left( \frac{\sin ax}{ax} \times a + b \right)x}{\left( a + \frac{\sin bx}{bx} \times b \right)x} \right]\]
\[ = \frac{1 \times a + b}{a + b}\]
\[ = 1\]
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