Question

Discuss the continuity or otherwise of the function f(x) defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space 2 straight x over denominator sin space 3 straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space space space 2 space space space space space comma space straight x equals 0 end cell end table close at space straight x equals 0$

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin space 2 straight x over denominator sin space 3 straight x end fraction comma space straight x not equal to 0 end cell row cell space space space space space space 2 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 sin space 2 straight x over denominator sin space 3 straight x end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator begin display style fraction numerator sin space 2. straight x over denominator 2 straight x end fraction end style cross times 2 over denominator begin display style fraction numerator sin space 3 straight x over denominator 3 straight x end fraction end style cross times 3 end fraction equals 2 over 3 cross times 1 over 1 equals 2 over 3 But space straight f left parenthesis 0 right parenthesis equals 2 therefore space 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 space space space space space space space rightwards double arrow space straight f space is space continous space at space straight x equals 0

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