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JEE Main & Advanced Mathematics Functions Question Bank Continuity

  • question_answer
    If \[f(x)=\left\{ \begin{align}   & \sin x,\ x\ne n\pi ,\ \ n\in Z \\  & \,\,\,\,\,\,2,\,\text{otherwise} \\ \end{align} \right.\] and \[g(x)=\left\{ \begin{align}   & {{x}^{2}}+1,\ x\ne 0,\,2 \\  & \,\,\,\,\,\,\,\,\,4,\,x=0 \\  & \,\,\,\,\,\,\,\,\,\,5,x=2 \\ \end{align} \right.,\]  then \[\underset{x\to 0}{\mathop{\lim }}\,g\,\{f(x)\}\] is [Kurukshetra CEE 1996]

    A)            5

    B)            6

    C)            7

    D)            1

    Correct Answer: D

    Solution :

               As we are given \[f(x)=\sin x\], if \[x\ne n\pi \]                    i.e., \[x\ne 0,\,\pi ,\,\,2\pi ,....\]= 2 otherwise                    \[\therefore \,\,\,\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,g\,\{f(x)\}=\,\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,g\,\{\sin x\}=\,\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,({{\sin }^{2}}x+1)\] = 1                    Similarly, \[\,\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,g\,\left\{ f(x) \right\}=1.\]


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