Question

Discuss continuity of function at x = 2, where
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 2 straight x squared 3 straight x minus 2 over denominator straight x minus 2 end fraction comma space straight x not equal to 2 end cell row cell space space space space space space space space space 3 space space space space space space space space space comma space straight x equals 2 end cell end table close$

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator 2 straight x squared 3 straight x minus 2 over denominator straight x minus 2 end fraction comma space straight x not equal to 2 end cell row cell space space space space space space space space space 3 space space space space space space space space space comma space straight x equals 2 end cell end table close space Lt with straight x rightwards arrow 2 below space straight f left parenthesis straight x right parenthesis space equals space Lt with straight x rightwards arrow 2 below space fraction numerator 2 straight x squared minus 3 straight x minus 2 over denominator straight x minus 2 end fraction equals space Lt with straight x rightwards arrow 2 below fraction numerator left parenthesis straight x minus 2 right parenthesis left parenthesis 2 straight x plus 2 right parenthesis over denominator straight x minus 2 end fraction equals space Lt with straight x rightwards arrow 2 below space left parenthesis 2 straight x plus 1 right parenthesis space space space space space space space space space space space space space space space space space equals 4 plus 1 equals 5 Also space straight f left parenthesis 2 right parenthesis equals 3 therefore space Lt with straight x rightwards arrow 2 below space straight f left parenthesis straight x right parenthesis not equal to straight f left parenthesis 2 right parenthesis

∴ f(x) is discontinuous at x = 2.

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