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`k in (-oo, 0)``k in (1, oo)``k in (-1, oo)` None of these

Answer :

DSolution :

If f(x) is continuous at x = 0, then <br> `lim_(x to 0^(-))f(x)=lim_(x to 0^(+))f(x)=f(0)=0` <br> So, `lim_(x to 0^(-)) f(x)=lim_(h to 0)f(0-h)` <br> `=lim_(h to 0)(-1)^(k)h^(k)sin(-(1)/(h))` <br> `=-lim_(h to 0)(-1)^(k)(h)^(k)*sin((1)/(h))=0 " "[because " only for " k gt 0]` <br> `and lim_(x to 0^(+))f(x)=lim_(h to 0)f(0+h)=lim_(h to 0) h^(k)sin((1)/(h))=0. " " [because " only for " k gt 0]` <br> So, f(x) is continuous at x = 0, if ` k gt 0`. <br> So, `k in (0, oo)` is the answer.