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Question
\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{a^{mx} - b^{nx}}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{a^{mx} - 1 - b^{nx} + 1}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\left( a^{mx} - 1 \right) - \left( b^{nx} - 1 \right)}{x} \right]\]
\[ = \lim_{x \to 0} \left[ \left( \frac{a^{mx} - 1}{mx} \right) \times m - \left( \frac{b^{nx} - 1}{nx} \right) \times n \right]\]
\[ = m \log a - n \log b\]
\[ = \log \left( a \right)^m - \log \left( b \right)^n \]
\[ = \log \left( \frac{a^m}{b^n} \right)\]
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