A) 0
B) 3
C) 2
D) 1
Correct Answer: D
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\frac{d}{dx}\int\limits_{0}^{{{x}^{2}}}{{{\sec }^{2}}tdt}}{\frac{d}{dx}\left( x\sin x \right)}=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sec }^{2}}{{x}^{2}}.2x}{\sin x+x\cos x}\] (by L? Hospital rule) \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2{{\sec }^{2}}{{x}^{2}}}{\left( \frac{\sin x}{x}+\cos x \right)}=\frac{2\times 1}{1+1}=1\]
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