Question

Given that
$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell fraction numerator 1 minus cos space 4 straight x over denominator straight x squared end fraction space space space space space space space space space space comma space if space straight x less than 0 end cell row cell space space space space space space straight a space space space space space space space space space space space space space space space space space space space space space space space comma space if space straight x equals 0 end cell row cell fraction numerator square root of straight x over denominator square root of 16 plus square root of straight x end root minus 4 end fraction space space comma space if space straight x greater than 0 end cell end table close$
If f(x) is continous at x=0, find the value of a.

Solution

We space have straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell fraction numerator 1 minus cos space 4 straight x over denominator straight x squared end fraction space space space space space space space space space space comma space if space straight x less than 0 end cell row cell space space space space space space straight a space space space space space space space space space space space space space space space space space space space space space space space comma space if space straight x equals 0 end cell row cell fraction numerator square root of straight x over denominator square root of 16 plus square root of straight x end root minus 4 end fraction space space comma space if space straight x greater than 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 Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator 1 minus cos space 4 straight x over denominator straight x squared end fraction space space space space space space space space space space left square bracket Put space straight x equals 0 minus straight h space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 0 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 Lt with straight h rightwards arrow 0 below fraction numerator 1 minus cos left parenthesis negative 4 straight h right parenthesis over denominator left parenthesis negative straight h right parenthesis squared end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator 1 minus cos space 4 straight h over denominator straight h squared end fraction space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator 2 sin squared 2 straight h over denominator straight h squared end fraction equals 8 space Lt with straight h rightwards arrow 0 below open parentheses fraction numerator sin space 2 straight h over denominator 2 straight h end fraction close parentheses squared equals 8 left parenthesis 1 right parenthesis squared equals 8 space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator square root of straight x over denominator square root of 16 plus square root of straight x end root minus 4 end fraction equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator square root of straight x over denominator square root of 16 plus square root of straight x end root minus 4 end fraction cross times fraction numerator square root of 16 plus square root of straight x end root plus 4 over denominator square root of 16 plus square root of straight x end root plus 4 end fraction space space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator square root of straight x open parentheses square root of 16 plus square root of straight x end root minus 4 close parentheses over denominator left parenthesis 16 plus square root of straight x right parenthesis minus 16 end fraction equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator square root of straight x open parentheses square root of 16 plus square root of straight x end root plus 4 close parentheses over denominator square root of straight x end fraction space space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 to the power of plus below open parentheses square root of 16 plus square root of straight x end root minus 4 close parentheses 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 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 space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow 0 to the power of plus right square bracket space space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow 0 to the power of plus below open parentheses square root of 16 plus square root of straight x end root minus 4 close parentheses equals square root of 16 plus 0 end root plus 4 equals 4 plus 4 equals 8

Also f(0) = a
Since f(x) is continuous at x = 0
$therefore space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis therefore space space space space space space space space space space space 8 equals 8 equals straight a rightwards double arrow space space space space space space space space space space straight a equals 8$

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