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