Evaluate the following limit: _ x 0 a^mx-b^nx - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{mx}-\text{b}^\text{nx}}{\sin\text{kx}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{mx}-\text{b}^\text{nx}}{\sin\text{kx}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{mx}-\text{b}^\text{nx}}{\text{kx}\frac{\sin\text{kx}}{\text{kx}}}$
$=\frac{1}{\text{x}}\lim\limits_{\text{x}\rightarrow0}\frac{\frac{\big(\text{a}^\text{mx}-\text{b}^\text{nx}\big)}{\text{x}}}{\frac{\sin\text{kx}}{\text{kx}}}$
$=\frac{1}{\text{x}}\text{log}\frac{\text{a}^\text{m}}{\text{b}^\text{n}}$

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