mathop {lim }limits_{x to 0} left( {frac{{{a^ | vedclass

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Question

(B) We are given the limit: $\mathop {\lim }\limits_{x 	o 0} \frac{{{a^x} - {b^x}}}{x}$.To solve this,we add and subtract $1$ in the numerator:$\math

Vedclass · Accepted Answer

$\log \left( {\frac{a}{b}} \right)$

Vedclass · Answer

(B) We are given the limit: $\mathop {\lim }\limits_{x 	o 0} \frac{{{a^x} - {b^x}}}{x}$.To solve this,we add and subtract $1$ in the numerator:$\mathop {\lim }\limits_{x 	o 0} \frac{{{a^x} - 1 - ({b^x} - 1)}}{x} = \mathop {\lim }\limits_{x 	o 0} \left( \frac{{{a^x} - 1}}{x} - \frac{{{b^x} - 1}}{x} ight)$.Using the standard limit formula $\mathop {\lim }\limits_{x 	o 0} \frac{{{k^x} - 1}}{x} = \log k$:$= \log a - \log b$.Using the logarithmic property $\log m - \log n = \log \left( \frac{m}{n} ight)$:$= \log \left( \frac{a}{b} ight)$.

$\mathop {\lim }\limits_{x \to 0} \left( {\frac{{{a^x} - {b^x}}}{x}} \right) = $

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