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Question
\[\lim_{x \to 0} \frac{\log \left( a + x \right) - \log \left( a - x \right)}{x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{\log \left( a + x \right) - \log \left( a - x \right)}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\log \left( \frac{a + x}{a - x} \right)}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\log \left( 1 + \frac{a + x}{a - x} - 1 \right)}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\log \left( 1 + \frac{a + x - a + x}{a - x} \right)}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\log \left( 1 + \frac{2x}{a - x} \right)}{\frac{2x}{a - x} \times \left( \frac{a - x}{2} \right)} \right]\]
\[x \to 0\]
\[ \therefore \frac{2x}{a - x} \to 0\]
\[Let y = \frac{2x}{a - x}\]
\[ = \lim_{y \to 0} \left[ \frac{\log \left( 1 + y \right)}{y} \right] \times \lim_{x \to 0} \left( \frac{1}{\frac{a - x}{2}} \right)\]
\[ = 1 \times \frac{2}{a}\]
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