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11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-6

  • question_answer
    The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\mathbf{x}}^{\mathbf{7}}}-\mathbf{2}{{\mathbf{x}}^{\mathbf{5}}}+\mathbf{1}}{{{x}^{3}}-3{{x}^{2}}+2}\]is equal to

    A)  0                   

    B)  1              

    C)  ¥                                

    D)  \[-1\]

    Correct Answer: B

    Solution :

    [b] \[\because \underset{x\to 1}{\mathop{\lim }}\,\frac{{{x}^{7}}-2{{x}^{5}}+1}{{{x}^{3}}-3{{x}^{2}}+2}\] which is \[\frac{0}{0}\]form  \[=\underset{x\to 1}{\mathop{\lim }}\,\frac{\frac{d}{dx}\left( {{x}^{7}}-2{{x}^{5}}+1 \right)}{\frac{d}{dx}\left( {{x}^{3}}-3{{x}^{2}}+2 \right)}\] [By L. Hospital Rule] \[=\underset{x\to 1}{\mathop{\lim }}\,\frac{7{{x}^{6}}-10{{x}^{4}}}{3{{x}^{2}}-6x}=\frac{7-10}{3-6}=\frac{-3}{-3}=1\] Hence, option [b]is correct.


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