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