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KVPY Sample Paper KVPY Stream-SX Model Paper-11

  • question_answer
    For each \[x\in R\] let \[[x]\] be the greatest integer less than or equal to\[x\]. then Then \[\underset{x\to {{0}^{+}}}{\mathop{\text{lim}}}\,\frac{x([x])+\left| x \right|\sin [x]}{\left| x \right|}\] is equal  to:

    A) \[-\,\sin 1\]

    B) 0

    C) 1

    D) sin1

    Correct Answer: A

    Solution :

    \[\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,\frac{x\,([x]+\left| x \right|)sin[x]}{\left| x \right|}\] \[x\to 0\,-\] \[\left. \begin{align}   & [x]=-1 \\  & \left| x \right|=-x \\ \end{align} \right\}\Rightarrow \underset{x\to \,0{{\,}^{-}}}{\mathop{\lim }}\,\frac{x\,(-x-1)\sin (-1)}{-x}=-\sin 1.\]


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