Let f:R to R be such that f(1) = 3 and f'(1) | vedclass

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(C) Let $L = \lim_{x 	o 0} \left\{ \frac{f(1 + x)}{f(1)} ight\}^{\frac{1}{x}}$.Taking natural logarithm on both sides,$\ln L = \lim_{x 	o 0} \frac

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$e^2$

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(C) Let $L = \lim_{x 	o 0} \left\{ \frac{f(1 + x)}{f(1)} ight\}^{\frac{1}{x}}$.Taking natural logarithm on both sides,$\ln L = \lim_{x 	o 0} \frac{1}{x} \ln \left( \frac{f(1 + x)}{f(1)} ight)$.Using the definition of the derivative or $L$'Hopital's rule,$\ln L = \lim_{x 	o 0} \frac{\ln f(1 + x) - \ln f(1)}{x}$.Applying $L$'Hopital's rule,$\ln L = \lim_{x 	o 0} \frac{f'(1 + x)}{f(1 + x)} = \frac{f'(1)}{f(1)}$.Given $f(1) = 3$ and $f'(1) = 6$,we have $\ln L = \frac{6}{3} = 2$.Therefore,$L = e^2$.

Let $f:R \to R$ be such that $f(1) = 3$ and $f'(1) = 6$. Then $\lim_{x \to 0} \left\{ \frac{f(1 + x)}{f(1)} \right\}^{\frac{1}{x}}$ equals

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