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