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

$(a)\;1\qquad(b)\;e^{1/2}\qquad(c)\;e^2\qquad(d)\;e^3$

$\lim\limits_{x\to 0}\big(\large\frac{f(1+x)}{f(1)}\big)^{1/x}$
$\Rightarrow e^{\log}\lim\limits_{x\to 0}\bigg[\large\frac{f(1+x)}{f(1)}\bigg]^{1/x}$
$\Rightarrow e^{\lim\limits_{x\to 0}\large\frac{1}{x}\log\big(\large\frac{f(1+x)}{f(1)}\big)}$
$\Rightarrow e^{\lim\limits_{x\to 0}\large\frac{f(1)}{f(1+x)}\frac{f'(1+x)}{f(1)}}$
$\Rightarrow \large\frac{f'(1)}{f(1)}$
$\Rightarrow e^{6/3}$
$\Rightarrow e^2$
Hence (c) is the correct answer.