GIven If f(x) is continuous at x = 0, find the value of k

Question

GIven $straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space 4 straight x over denominator 8 straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space space straight k space space space space space space space comma straight x equals 0 end cell end table close$
If f(x) is continuous at x = 0, find the value of k

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space 4 straight x over denominator 8 straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space space space space straight k space space space space space space space comma 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 1 minus cos space 4 straight x over denominator 8 straight x squared end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator 2 space sin squared 2 straight x over denominator 8 straight x squared end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator sin squared 2 straight x over denominator 4 straight x squared end fraction space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 below open parentheses fraction numerator sin space 2 straight x over denominator 2 straight x end fraction close parentheses squared equals left parenthesis 1 right parenthesis squared equals 1 SInce space straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 0. therefore space 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 space space space space space space space rightwards double arrow space straight k equals 1

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