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JEE Main & Advanced Mathematics Functions Question Bank Critical Thinking

  • question_answer
    The \[\underset{x\to 0}{\mathop{\lim }}\,{{(\cos x)}^{\cot x}}\]is                                                [RPET 1999]

    A) -1

    B) 0

    C) 1

    D) None of these

    Correct Answer: C

    Solution :

    • \[y=\underset{x\to 0}{\mathop{\lim }}\,{{(\cos x)}^{\cot x}}\]                   
    • Taking log on both sides,           
    • Þ \[\log y=\underset{x\to 0}{\mathop{\lim }}\,\,\,\cot x\log \cos x\]           
    • Þ \[\log y=\underset{x\to 0}{\mathop{\lim }}\,\frac{\log \cos x}{\tan x}\], \[\left( \frac{0}{0}\text{form} \right)\]           
    • Applying L-Hospital?s rule,                   
    • Þ \[\log y=\underset{x\to 0}{\mathop{\lim }}\,\frac{-\tan x}{{{\sec }^{2}}x}\]= 0                   
    • Þ \[y={{e}^{0}}\] Þ \[y=1\].


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