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JEE Main & Advanced Mathematics Functions Question Bank Critical Thinking

  • question_answer
    The integer n for which \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{(\cos x-1)\,(\cos x-{{e}^{x}})}{{{x}^{n}}}\] is a finite non-zero number is [IIT Screening 2002]

    A) 1

    B) 2

    C) 3

    D) 4

    Correct Answer: C

    Solution :

    • n cannot be negative integer for then the limit = 0           
    • Limit \[=\underset{x\to 0}{\mathop{\lim }}\,\frac{2{{\sin }^{2}}\frac{x}{2}}{{{2}^{2}}{{(x/2)}^{2}}}\frac{{{e}^{x}}-\cos x}{{{x}^{n-2}}}=\frac{1}{2}\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}-\cos x}{{{x}^{n-2}}}\]           
    • \[(n\ne 1\] for then the limit \[=0)\]           
    • \[=\frac{1}{2}\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}+\sin x}{(n-2){{x}^{n-3}}}\].           
    • So, if \[n=3,\] the limit is \[\frac{1}{2(n-2)}\] which is finite.            
    • If \[n=4,\] the limit is infinite.


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