Question

Show that the function is continuous at x = 3.
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 7 straight x plus 12 over denominator straight x minus 3 end fraction comma space straight x space not equal to 3 end cell row cell space space space space space space space space space space minus 1 space space space space space space space space comma space straight x equals 3 end cell end table close$

Solution

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 7 straight x plus 12 over denominator straight x minus 3 end fraction comma space straight x space not equal to 3 end cell row cell space space space space space space space space space space minus 1 space space space space space space space space comma space straight x equals 3 end cell end table close space Lt with straight x rightwards arrow 3 below space straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 3 below space fraction numerator straight x squared minus 7 straight x plus 12 over denominator straight x minus 3 end fraction equals space Lt with straight x rightwards arrow 3 below space fraction numerator left parenthesis straight x minus 3 right parenthesis left parenthesis straight x minus 4 right parenthesis over denominator straight x minus 3 end fraction equals space Lt with straight x rightwards arrow 3 below space left parenthesis straight x minus 4 right parenthesis equals 3 minus 4 equals negative 1 Also space straight f left parenthesis 3 right parenthesis equals negative 1 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 negative 1 comma space which space is space infinite

∴ f(x) is continuous at x = 3.

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