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JEE Main & Advanced Mathematics Functions Question Bank Critical Thinking

  • question_answer
     If [.] denotes the greatest integer less than  or equal to x, then the value of \[\underset{x\to 1}{\mathop{\lim }}\,(1-x+[x-1]+[1-x])\]is

    A) 0

    B) 1

    C) -1

    D) None of these

    Correct Answer: C

    Solution :

    • We have \[\underset{x\to 1-}{\mathop{\lim }}\,\,\,(1-x+[x-1]+[1-x])\]                           
    • \[=\underset{h\to 0}{\mathop{\lim }}\,\,\,(1-(1-h)+[1-h-1]+[1-(1-h)])\,\]                           
    • \[=\underset{h\to 0}{\mathop{\lim }}\,\,(h+[-h]+[h])=\underset{h\to 0}{\mathop{\lim }}\,\,(h-1+0)=-1\]      
    • and \[\underset{x\to 1+}{\mathop{\lim }}\,\,(1-x+[x-1]+[1-x])\,\]           
    • \[=\underset{h\to 0}{\mathop{\lim }}\,\,(1-(1+h)+[1+h-1]+[1-(1+h)])\]           
    • \[=\underset{h\to 0}{\mathop{\lim }}\,\,(-h+[h]+[-h])=\underset{h\to 0}{\mathop{\lim }}\,\,(-h+0-1)=-1\]           
    • \ \[\underset{x\to 1}{\mathop{\lim }}\,\,f(x)=-1\].


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