Question

Examine the continuity of
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction comma space if space straight x not equal to 0 end cell row cell space space space space 0 space space space comma space if space straight x equals 0 end cell end table close space space space space space space space space space at space straight x equals 0$

Solution

Here space straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction comma space if space straight x not equal to 0 end cell row cell space space space space 0 space space space comma space if space straight x equals 0 end cell end table close space Lt with straight x rightwards arrow 0 to the power of minus below space straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator straight x over denominator open vertical bar straight x close vertical bar end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket Put space straight x equals 0 plus straight h comma space straight h greater than 0 right square bracket space space space space space space space space space space 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 to the power of minus below fraction numerator 0 plus straight h over denominator open vertical bar 0 plus straight h close vertical bar end fraction equals space Lt with straight h rightwards arrow 0 below fraction numerator straight h over denominator open vertical bar straight h close vertical bar end fraction equals space Lt with straight h rightwards arrow 0 below space straight h over straight h space equals space Lt with straight h rightwards arrow 0 below space space space 1 equals 1 therefore space Lt with straight x rightwards arrow 0 to the power of minus below space straight f left parenthesis straight x right parenthesis space not equal to space Lt with straight x rightwards arrow 0 to the power of plus below therefore straight f space is space continous space at space straight x equals 0. space space space space space space space

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