Question

$If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x space sin 1 over straight x comma space space space straight x not equal to 0 end cell row cell 0 space space space space space space space space space comma space space space straight x equals 0 end cell end table close the space show space that space straight f space is space not space differentiable space at space straight x equals 0.$

Solution

Here space If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x space sin 1 over straight x comma space space space straight x not equal to 0 end cell row cell 0 space space space space space space space space space comma space space space straight x equals 0 end cell end table close straight R. straight H. straight D equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 0 right parenthesis over denominator straight x minus 0 end fraction equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator straight x space sin begin display style 1 over straight x end style minus 0 over denominator straight x end fraction equals Lt with straight x rightwards arrow 0 to the power of plus below sin 1 over straight x comma space which space does space not space exist therefore space straight f space is space not space differentiable space at space straight x equals 0

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