Question

Determine the value of the constant k so that the function f defined as
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator tan space 2 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space straight k space space space space space space comma space straight x equals 0 end cell end table close is space continous space at space straight x equals 0$

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 tan space 2 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space straight k space space space space space space comma space straight x equals 0 end cell end table close 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 tan space 2 straight x over denominator straight x end fraction equals 2 space Lt with straight x rightwards arrow 0 below fraction numerator tan space 2 straight x space over denominator 2 straight x end fraction equals 2 cross times 1 equals 2 Since space straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 0. space therefore straight f left parenthesis 0 right parenthesis equals Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis space space space space space space space space space space space space space space space space space space space rightwards double arrow space straight k equals 2.

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