Determine the value of constant A so that the functionis continous at x=0

Question

Determine the value of constant A so that the function
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos space straight x over denominator straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space space space space straight A space space space space space space comma space straight x equals 0 end cell end table close$
is continous at x=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 1 minus cos space straight x over denominator 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 straight A 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 1 minus cos space straight x over denominator straight x squared end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator 2 sin space straight x squared begin display style straight x over 2 end style over denominator straight x squared end fraction equals 1 half Lt with straight x rightwards arrow 0 below open parentheses fraction numerator sin begin display style straight x over 2 end style over denominator begin display style straight x over 2 end style end fraction close parentheses squared equals 1 half cross times 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 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 A equals 1 half.

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