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Find the value of p for which the function $f(x)=\left\{\begin{array}{ 1 1}\large\frac{(4^x-1)^3}{\sin\big(x/p\big)\log \big(1+\Large\frac{x^2}{3}\big)}&x\neq 0\\12(\log 4)^3&x=0\end{array}\right.$ is continuous at $x=0$

$(a)\;1\qquad(b)\;4\qquad(c)\;3\qquad(d)\;2$

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$f$ continuous at $x=0$
$\Rightarrow f(0)=\lim\limits_{x\to 0}f(x)$
$\Rightarrow 12(\log 4)^3=\lim\limits_{x\to 0}\big[\large\frac{4^x-1}{x}\big]^3.\frac{x^3}{\Large\frac{\sin\big(x/p\big)}{x/p}\big(\Large\frac{x}{p}\big)}\times \large\frac{1}{\Large\frac{x^2}{3}\log[1+(x^2/3)]^{1/x^3}}$
$\Rightarrow 12(\log 4)^3$
$\Rightarrow (\log 4)^3.3p.\large\frac{1}{\log e}$
$\Rightarrow 3p(\log 4)^3$
$\Rightarrow 3p=12$
$\Rightarrow p=4$
Hence (b) is the correct answer.
answered Dec 31, 2013 by sreemathi.v
 

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