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If \mathrm{f(x)=\left\{\begin{array}{cc} x^a \cdot \sin \left(\frac{1}{x}\right), & x \neq 1 \\ 0 & , \quad x=0 \end{array}\right.} is continuous but non-differentiable at x = 0, then 

Option: 1

\mathrm{a \in(-1,0)}


Option: 2

a \in(0,2)


Option: 3

a \in(0,1]


Option: 4

\mathrm{a \in[1,2)}


Answers (1)

best_answer

\mathrm{f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=\lim _{h \rightarrow 0} \frac{h^a \sin \left(\frac{1}{h}\right)}{h}=\lim _{h \rightarrow 0} h^{a-1} \cdot \sin \left(\frac{1}{h}\right)}

This limit will not exist if \mathrm{a-1 \leq 0 \Rightarrow a \leq 1}

Posted by

Ajit Kumar Dubey

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