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Question
\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{\sin kx}\]
Solution
\[\lim_{x \to 0} \left[ \frac{a^{mx} - b^{nx}}{\sin \left( kx \right)} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\left( a^{mx} - 1 \right) - \left( b^{nx} - 1 \right)}{\sin kx} \right]\]
\[\text{ Dividing the numerator and the denomiantor by } x:\]
\[ = \lim_{x \to 0} \left[ \frac{m\left( \frac{a^{mx} - 1}{mx} \right) - n\left( \frac{b^{nx} - 1}{nx} \right)}{k \times \frac{\sin kx}{kx}} \right]\]
\[ = \frac{\left( m \log a - n \log b \right)}{k \times 1}\]
\[ = \frac{1}{k} \left[ \log \left( a \right)^m - \log \left( b \right)^n \right]\]
\[ = \frac{1}{k} \log \left( \frac{a^m}{b^n} \right)\]
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