Evaluate the following limit: _ x 0 a^mx-1 b^nx-1 ,n =0 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{mx}-1}{\text{b}^\text{nx}-1},\text{n}\not=0$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{mx}-1}{\text{b}^\text{nx}-1},\text{n}\not=0$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{mx}-1}{\text{mx}}\times\frac{1}{\lim\limits_{\text{x}\rightarrow0}\frac{\text{b}^\text{nx}-1}{\text{nx}}}\times\frac{\text{m}}{\text{n}}$
$=\frac{\text{mloga}}{\text{nlogb}},\text{n}\not=0$

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