Let f : R R be such that f ( 1 ) = 3 and f ' ( 1 ) = 6 . Then lim x 0 ( f ( 1 + x ) f ( 1 ) ) 1 / x equals [2002]

Question

Let $f : R \to R$ be such that $f (1) = 3$ and $f' (1) = 6$ . Then $\lim_{x \to 0} {(\frac{f (1 + x)}{f (1)})}^{1 / x}$ equals [2002]

Smart Achievers · Accepted Answer

(3)

Given $f : R \to R,$ $f (1) = 3$ and $f' (1) = 6$

Then, $\lim_{x \to 0} {[\frac{f (1 + x)}{f (1)}]}^{1 / x}$

$= e^{\lim_{x \to 0} \frac{1}{x} [\log f (1 + x) - \log f (1)]}$

$= e^{\lim_{x \to 0} \frac{\frac{1}{f (1 + x)} f' (1 + x)}{1}}$ (Using L'Hospital rule)

$= e^{\frac{f' (1)}{f (1)}} = e^{6 / 3} = e^{2}$

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