Evaluate the following limit: _ x 0 a^x+b^x-2 x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}+\text{b}^\text{x}-2}{\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}+\text{b}^\text{x}-2}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{a}^\text{x}-1}{\text{x}}+\lim\limits_{\text{x}\rightarrow0}\frac{\text{b}^\text{x}-1}{\text{x}}$
$=\text{loga}+\text{logb}$
$=\text{log (ab)}$

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