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Continuity And Differentiability

Question
CBSEENMA12034542
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 Determine the value of a, by c for which the functions
straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell fraction numerator sin left parenthesis straight a plus 1 right parenthesis straight x plus sin space straight x over denominator straight x end fraction space space space comma space straight x less than 0 end cell row cell space space space space space space space space space space space space space space space space space space space space straight c space space space space space space space space space space space space space space space comma space straight x equals 0 end cell row cell fraction numerator square root of straight x plus bx squared end root minus square root of straight x over denominator bx to the power of begin display style 3 over 2 end style end exponent end fraction space space space space space space space comma space straight x greater than 0 end cell end table close
is continous at x=0.

Solution
We space have straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell fraction numerator sin left parenthesis straight a plus 1 right parenthesis straight x plus sin space straight x over denominator straight x end fraction space space space comma space straight x less than 0 end cell row cell space space space space space space space space space space space space space space space space space space space space straight c space space space space space space space space space space space space space space space comma space straight x equals 0 end cell row cell fraction numerator square root of straight x plus bx squared end root minus square root of straight x over denominator bx to the power of begin display style begin inline style bevelled 3 over 2 end style end style end exponent end fraction space space space space space space space comma 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 sin left parenthesis straight a plus 1 right parenthesis straight x plus sin space straight x over denominator straight x end fraction equals Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator sin left parenthesis straight a plus 1 right parenthesis straight x over denominator straight x end fraction plus space Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator sin space straight x over denominator 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 minus below fraction numerator sin left parenthesis straight a plus 1 right parenthesis straight x over denominator left parenthesis straight a plus 1 right parenthesis straight x end fraction. left parenthesis straight a plus 1 right parenthesis plus space Lt with straight x rightwards arrow 0 to the power of minus below fraction numerator sin space straight x over denominator straight x end fraction equals 1. left parenthesis straight a plus 1 right parenthesis plus 1 equals straight a plus 2 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 plus bx squared end root minus square root of straight x over denominator bx to the power of begin display style bevelled 3 over 2 end style end exponent 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 plus bx squared end root minus square root of straight x over denominator bx to the power of bevelled 3 over 2 end exponent end fraction cross times fraction numerator square root of straight x plus bx squared end root plus square root of straight x over denominator square root of straight x plus bx squared end root plus 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 fraction numerator left parenthesis straight x plus bx squared right parenthesis minus straight x over denominator bx to the power of bevelled 3 over 2 end exponent open parentheses square root of straight x plus bx squared end root minus square root of straight x close parentheses end fraction equals Lt with straight x rightwards arrow 0 to the power of plus below fraction numerator bx squared over denominator bx to the power of bevelled 3 over 2 end exponent open parentheses square root of straight x plus bx squared end root minus square root of straight x close parentheses 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 below fraction numerator bx squared over denominator bx squared open parentheses square root of 1 plus bx end root plus 1 close parentheses end fraction equals Lt with straight x rightwards arrow 0 below fraction numerator 1 over denominator square root of straight x plus bx plus 1 end root end fraction equals fraction numerator 1 over denominator square root of straight x plus 0 end root plus 1 end fraction equals 1 half.
Also f(0) = c
Since f 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 rightwards double arrow space straight a plus 2 equals 1 half equals straight c space rightwards double arrow space straight a equals negative 3 over 2 comma straight c equals 1 half
b can be any real number except 0.
 

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