Question

Examine the continuity of f (x) at x = 0.
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator straight x over denominator 2 open vertical bar straight x close vertical bar end fraction comma space straight x not equal to 0 end cell row cell space space space space space 1 half comma space straight x equals 0 end cell end table close$

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Solution

Here 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 over denominator 2 open vertical bar straight x close vertical bar end fraction comma space straight x not equal to 0 end cell row cell space space space space space 1 half comma space straight x equals 0 end cell end table close space Lt with straight x rightwards arrow 0 to the power of minus below 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 2 open vertical bar straight x close vertical bar end fraction equals space Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator straight x over denominator 2 left parenthesis negative straight x right parenthesis end fraction space space space left square bracket because space open vertical bar straight x close vertical bar equals negative straight x space for space straight x less than 0 right square bracket space space space space space space space space space space space space space space space space equals space minus 1 half space Lt with straight x rightwards arrow 0 to the power of minus below equals negative 1 half space Lt with straight x rightwards arrow 0 to the power of minus below space left parenthesis 1 right parenthesis equals negative 1 half cross times 1 equals space minus 1 half space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator straight x over denominator 2 open vertical bar straight x close vertical bar end fraction equals space Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator straight x over denominator 2 space straight x end fraction space space space space space space space space space left square bracket because space open vertical bar straight x close vertical bar equals straight x space for space straight x greater than 0 right square bracket space space space space space space space space space space space space space space space space equals 1 half Lt with straight x rightwards arrow 0 to the power of plus below left parenthesis 1 right parenthesis equals 1 half cross times 1 equals 1 half therefore space space space space space space space Lt with straight x rightwards arrow 0 to the power of minus below 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 straight f left parenthesis straight x right parenthesis space space space space space space space space space rightwards double arrow space space space straight f left parenthesis straight x right parenthesis space is space discontinous space at space straight x equals 0.

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