KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    If \[f(x)=\left\{ \begin{matrix}    \frac{[x]-1}{x-1}, & x\ne 1  \\    0, & x=1  \\ \end{matrix} \right.\] then f(x) is

    A) Continuous as well as differentiable at \[x=1\]

    B) Differentiable but not continuous at \[x=1\]

    C) Continuous but not differentiable at \[x=1\]

    D) Neither continuous nor differentiable at\[x=1\]

    Correct Answer: D

    Solution :

    we have \[f(x)=\left\{ \begin{align}   & \begin{matrix}    \frac{-1}{x-1}, & 0<x<1  \\    \frac{1-1}{x-1}, & 1<x<21  \\ \end{matrix} \\  & \begin{matrix}    0, & x=1  \\ \end{matrix} \\ \end{align} \right.\]
    \[{}^{\underset{h\to 0}{\mathop{\lim }}\,f(1-h)}=\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{(1-h)-1}\]
    \[=\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}=\infty \]
    \[\therefore \]\[f\left( x \right)\] is not continuous and hence not differentiable at x=1.

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