lim _{x rightarrow 0} frac{9^x-4^x}{x(9^x+4^x | vedclass

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Question

(A) We evaluate the limit: $\lim _{x \rightarrow 0} \frac{9^x-4^x}{x(9^x+4^x)}$Using the standard limit formula $\lim _{x \rightarrow 0} \frac{a^x-1}{

Vedclass · Accepted Answer

$\log \left(\frac{3}{2}\right)$

Vedclass · Answer

(A) We evaluate the limit: $\lim _{x \rightarrow 0} \frac{9^x-4^x}{x(9^x+4^x)}$Using the standard limit formula $\lim _{x \rightarrow 0} \frac{a^x-1}{x} = \log(a)$:$= \lim _{x \rightarrow 0} \left( \frac{9^x-1 - (4^x-1)}{x} \right) \cdot \frac{1}{9^x+4^x}$$= \left( \lim _{x \rightarrow 0} \frac{9^x-1}{x} - \lim _{x \rightarrow 0} \frac{4^x-1}{x} \right) \cdot \lim _{x \rightarrow 0} \frac{1}{9^x+4^x}$$= (\log(9) - \log(4)) \cdot \frac{1}{1+1}$$= \log \left( \frac{9}{4} \right) \cdot \frac{1}{2}$$= \log \left( \left( \frac{3}{2} \right)^2 \right) \cdot \frac{1}{2}$$= 2 \log \left( \frac{3}{2} \right) \cdot \frac{1}{2} = \log \left( \frac{3}{2} \right)$

$\lim _{x \rightarrow 0} \frac{9^x-4^x}{x(9^x+4^x)} = $

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