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The value of $\lim\limits_{x\to 0}\large\frac{\int\limits_0^{x^2} \sec^2tdt}{x\sin x}$ is

$(a)\;0\qquad(b)\;3\qquad(c)\;2\qquad(d)\;1$

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$\lim\limits_{x\to 0}\large\frac{\Large\frac{d}{dx}\int\limits_0^{x^2}\sec^2tdt}{\Large\frac{d}{dx}(x\sin x)}=\lim\limits_{x\to 0}\large\frac{\sec^2x^2.2x}{\sin x+x\cos x}$
$\lim\limits_{x\to 0}\large\frac{2\sec^2x^2}{(\sin x/x)+\cos x}=\large\frac{2\times 1}{1+1}$
$\Rightarrow \large\frac{2}{2}=$$1$
Hence (d) is the correct answer.
answered Dec 23, 2013 by sreemathi.v
 

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