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Question
\[\lim_{x \to 0} \frac{\log \left( 2 + x \right) + \log 0 . 5}{x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{\log \left( 2 + x \right) + \log \left( 0 . 5 \right)}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\log \left( \left( 2 + x \right) \times 0 . 5 \right)}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\log \left( 1 + \frac{x}{2} \right)}{\frac{x}{2} \times 2} \right]\]
\[ = \frac{1}{2} \times 1\]
\[ = \frac{1}{2}\]
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