Question

Discuss the continuity of the function

$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 1 minus cos 4 straight x over denominator straight x squared end fraction comma space straight x not equal to 0 end cell row cell space space space space space space space 8 space space space space comma space straight x equals 0 end cell end table close at space straight x equals 0$

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 1 minus cos 4 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 8 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 4. straight x over denominator straight x squared end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator 2 sin squared 2 straight x over denominator straight x squared end fraction equals 8 space 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 8 cross times left parenthesis 1 right parenthesis squared equals 8 Also space straight f left parenthesis 0 right parenthesis equals 8 therefore Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis space space space space space space space space space space rightwards double arrow straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 0.

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