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Question
\[\lim_{x \to 0} \frac{a^x - a^{- x}}{x}\]
Solution
\[ = \lim_{x \to 0} \left[ \frac{a^x - \frac{1}{a^x}}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{a^{2x} - 1}{a^x \cdot 2x} \right] \times 2\]
\[ = \lim_{x \to 0} \left[ \frac{a^{2x} - 1}{2x} \right] \times \frac{2}{a^x}\]
\[ = \lim_{x \to 0} \left[ \frac{a^{2x} - 1}{2x} \right] \times \frac{2}{a^x}\]
\[ = \log \left( a \right) \times \frac{2}{a^0}\]
\[ = 2 \log a\]
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