Question

find whether f is continuous at x = 3
$If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 9 over denominator straight x minus 3 end fraction comma space straight x not equal to 3 end cell row cell space space space space space space 6 space space space space space comma straight x equals 3 end cell end table close$

Solution

Here space space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 9 over denominator straight x minus 3 end fraction comma space straight x not equal to 3 end cell row cell space space space space space 6 space space space space space space comma straight x equals 3 end cell end table close Lt with straight x rightwards arrow 3 below space straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 3 below fraction numerator straight x squared minus 9 over denominator straight x minus 3 end fraction equals Lt with straight x rightwards arrow 3 below fraction numerator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x plus 3 right parenthesis over denominator straight x minus 3 end fraction Lt with straight x rightwards arrow 3 below left parenthesis straight x plus 3 right parenthesis space space space space space space space space space space space space space space space space space equals 3 plus 3 plus equals 6 Also space straight f left parenthesis 3 right parenthesis equals 6 therefore space Lt with straight x rightwards arrow 3 below space straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 3 right parenthesis equals 6 comma space which space is space finite

∴ f (x) is continuous at x = 3.

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