Question

For what value of k is the function

$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space 5 straight x over denominator 3 straight x end fraction comma space if space straight x not equal to 0 end cell row cell space space space space space space straight k space space space space space space comma space if space straight x equals 0 end cell end table close continous space at space straight x equals 0 ?$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space 5 straight x over denominator 3 straight x end fraction comma space if space straight x not equal to 0 end cell row cell space space space space space space straight k space space space space space space comma space if space straight x equals 0 end cell end table close space space Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 below fraction numerator sin space 5 straight x over denominator 3 straight x end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator sin space 5 straight x over denominator 5 straight x end fraction cross times 5 over 3 equals 5 over 3 space Lt with straight x rightwards arrow 0 below fraction numerator sin space 5 straight x over denominator 5 straight x end fraction equals 5 over 3 cross times 1 equals 5 over 3 Since space straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 0 therefore space straight f left parenthesis 0 right parenthesis equals Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis rightwards double arrow space space space space space straight k equals 5 over 3

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