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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{mx}-\text{b}^\text{nx}}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\text{m}\frac{\text{a}^\text{nx}-1}{\text{mx}}-\lim\limits_{\text{x}\rightarrow0}\text{n}\frac{\text{b}^\text{nx}-1}{\text{nx}}$
$=\text{mlog a}-\text{nlog b}$
$=\text{log}\Big(\frac{\text{a}^\text{m}}{\text{b}^\text{n}}\Big)$

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