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Question
\[\lim_{x \to 0} \frac{a^x + b^ x - c^x - d^x}{x}\]
Solution
\[\lim_{x \to 0} \left[ \frac{a^n + b^n - c^n - d^n}{x} \right]\]
\[ \lim_{x \to 0} \left[ \frac{a^n + b^n - 2 - c^n - d^n + 2}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\left( a^n - 1 \right) + \left( b^n - 1 \right) - \left( c^n - 1 \right) - \left( d^n - 1 \right)}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \left( \frac{a^n - 1}{x} \right) + \left( \frac{b^n - 1}{x} \right) - \left( \frac{c^n - 1}{x} \right) - \left( \frac{d^n - 1}{x} \right) \right]\]
\[ = \log a + \log b - \log c - \log d\]
\[ = \left( \log a + \log b \right) - \left( \log c + \log d \right)\]
\[ = \log \left( ab \right) - \log \left( cd \right)\]
\[ = \log \left( \frac{ab}{cd} \right)\]
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