Question

find whether f is continuous at x = 1.
$space If space straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x squared minus 1 over denominator straight x minus 1 end fraction comma space straight x not equal to 1 end cell row cell 2 space space space space space space space space space space space comma straight x equals 1 end cell end table close$

Solution

Here space 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 1 over denominator straight x minus 1 end fraction comma space straight x not equal to 1 end cell row cell 2 space space space space space space space space space space space comma straight x equals 1 end cell end table close therefore Lt with straight x rightwards arrow 1 below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 below begin inline style fraction numerator straight x squared minus 1 over denominator straight x minus 1 end fraction end style equals space Lt with straight x rightwards arrow 1 below fraction numerator left parenthesis straight x minus 1 right parenthesis left parenthesis straight x plus 1 right parenthesis over denominator straight x minus 1 end fraction equals Lt with straight x rightwards arrow 1 below left parenthesis straight x plus 1 right parenthesis equals 1 plus 1 therefore Lt with straight x rightwards arrow 1 below space straight f left parenthesis straight x right parenthesis equals 2 Also space straight f left parenthesis 1 right parenthesis equals 2 therefore space Lt with straight x rightwards arrow 1 below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 1 right parenthesis equals 2 comma space which space is space finite

∴ f is continuous at x = 1.

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