Question

Find A so that f(x) is continuous at x = 0.
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space 2 straight x over denominator 4 straight x squared end fraction comma space if space straight x not equal to 0 end cell row cell space space space space space space space space space space straight A space space space space space space space space comma space if space straight x equals 0 end cell end table close$

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 1 minus cos space 2 straight x over denominator 4 straight x squared end fraction comma space if space straight x not equal to 0 end cell row cell space space space space space space space space space space straight A space space 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 1 minus cos space 2 straight x over denominator 4 straight x squared end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator 2 sin squared space straight x over denominator 4 straight x squared end fraction equals 1 half Lt with straight x rightwards arrow 0 below open parentheses fraction numerator sin space straight x over denominator straight x end fraction close parentheses squared space space space space space space space space space space space space space space space equals 1 half left parenthesis 1 right parenthesis squared equals 1 half SInce space straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 0 therefore space space space straight f left parenthesis 0 right parenthesis space equals Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis space rightwards double arrow space straight A equals 1 half

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