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KVPY Sample Paper KVPY Stream-SX Model Paper-6

  • question_answer
    Evaluate\[\underset{x\,\to \,0}{\mathop{\lim }}\,\frac{x}{a}\,\,\left[ \frac{b}{x} \right],\] (where [.] denotes the greatest integer function).

    A) \[\frac{b}{a}\]                          

    B) 0

    C) \[\frac{a}{b}\]                          

    D) does not exist

    Correct Answer: A

    Solution :

    \[\underset{x\,\to \,0}{\mathop{\lim }}\,\,\,\frac{x}{a}\,\,\left[ \frac{b}{x} \right]=\underset{x\,\to \,0}{\mathop{\lim }}\,\frac{x}{a}\,\,\left( \frac{b}{x}-\left\{ \frac{b}{x} \right\} \right)\]
    \[=\frac{b}{a}\,\,\underset{x\,\to \,0}{\mathop{\lim }}\,\,\,\frac{x}{a}\,\,\left\{ \frac{b}{x} \right\}\]
    \[=\frac{b}{a}\] (\[\because \]\[\underset{x\,\to \,0}{\mathop{\lim }}\,\,\,\left\{ \frac{b}{x} \right\}\] is a finite number)


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