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JEE Main & Advanced Sample Paper JEE Main Sample Paper-47

  • question_answer
    The value of \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\int_{0}^{{{x}^{2}}}{\sin \,\sqrt{t}}}{{{x}^{3}}}dt\] is

    A)  0                                

    B)  2/9                

    C)  1/3                             

    D)  2/3

    Correct Answer: D

    Solution :

    \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\int_{0}^{{{x}^{2}}}{\sin \,\sqrt{t}}}{{{x}^{3}}}dt\] \[=\underset{x\to 0}{\mathop{\lim }}\,\,\frac{(isn\,x)\cdot 2x}{3{{x}^{2}}}\](by Leibnitz rule) \[=\frac{2}{3}\,\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\sin \,x}{x}=\frac{2}{3}\times 1=\frac{2}{3}\]


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