Question

Test the continuity of the function at x = 3, where
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x minus 3 close vertical bar over denominator straight x left parenthesis straight x minus 3 right parenthesis end fraction space space comma space space straight x not equal to 3 end cell row cell space space space space space space space 0 space space space space space space space space space comma space space straight x equals 3 end cell end table close$

Solution

We have

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator open vertical bar straight x minus 3 close vertical bar over denominator straight x left parenthesis straight x minus 3 right parenthesis end fraction space space comma space space straight x not equal to 3 end cell row cell space space space space space space 0 space space space space space space space space space space comma space space space straight x equals 3 end cell end table close space Lt with straight x rightwards arrow 3 to the power of minus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 3 to the power of minus below fraction numerator open vertical bar straight x minus 3 close vertical bar over denominator straight x left parenthesis straight x minus 3 right parenthesis end fraction space space space space space space space space space space space space left square bracket Put space straight x equals 3 minus straight h comma space straight h greater than 0 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 3 to the power of minus right square bracket space space space space space space space space space space space space space space space space equals space Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar 3 minus straight h minus 3 close vertical bar over denominator left parenthesis 3 minus straight h right parenthesis left parenthesis 3 minus straight h minus 3 right parenthesis end fraction space space space space space space space space space space space space space space space space equals space Lt with straight h rightwards arrow 0 below fraction numerator open vertical bar negative straight h close vertical bar over denominator left parenthesis 3 minus straight h right parenthesis left parenthesis negative straight h right parenthesis end fraction equals space Lt with straight h rightwards arrow 0 below fraction numerator straight h over denominator left parenthesis 3 minus straight h right parenthesis left parenthesis negative straight h right parenthesis end fraction space space space space space space space space space space space space space space space space equals space Lt with straight h rightwards arrow 0 below fraction numerator 1 over denominator straight h minus 3 end fraction equals fraction numerator 1 over denominator 0 minus 3 end fraction equals negative 1 third Also space straight f left parenthesis 3 right parenthesis space equals 0 therefore space Lt with straight x rightwards arrow 3 to the power of minus below space straight f left parenthesis straight x right parenthesis not equal to straight f left parenthesis 3 right parenthesis

∴ f(x) is discontinuous at x = 3.

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