Question

Discuss the continuity or otherwise of the function f(x) defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin 3 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space 1 space space space space space space comma space straight x equals 0 end cell end table close space at space straight x equals 0$

Solution

We space have space space space space space space space space space space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space 3 straight x over denominator straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space 1 space space space space space space comma 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 sin space 3 straight x over denominator straight x end fraction equals Lt with straight x rightwards arrow 0 below open square brackets 3. fraction numerator sin space 3 straight x over denominator 3 straight x end fraction close square brackets equals 3 Lt with straight x rightwards arrow 0 below fraction numerator sin space 3 straight x over denominator 3 straight x end fraction equals 3 cross times 1 equals 3 But space space straight f left parenthesis 0 right parenthesis equals 1 therefore Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis not equal to straight f left parenthesis 0 right parenthesis

⇒ f is not continuous at x = 0

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