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