Question

$If space space straight f open parentheses straight x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin open parentheses straight a plus 1 close parentheses plus 2 sinx over denominator straight x end fraction comma space straight x less than 0 end cell row cell 2 space space space space space space space space space space space space space space space space space space space space space space comma space x equals 0 end cell row cell fraction numerator square root of 1 plus b x end root minus 1 over denominator straight x end fraction space space space space space space comma space straight x greater than 0 end cell end table close$
is continuous at x = 0, then find the values of a and b.

Solution

Given space that space straight f space is space continous space at space straight x space equals space 0 If space space straight f open parentheses straight x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator sin open parentheses straight a plus 1 close parentheses plus 2 sinx over denominator straight x end fraction comma space straight x space less than space 0 end cell row cell 2 space space space space space space space space space space space space space space space space space space space space space space comma space x equals 0 end cell row cell fraction numerator square root of 1 plus b x end root minus 1 over denominator straight x end fraction space space space space space space comma space straight x space greater than space 0 end cell end table close Since space straight f left parenthesis straight x right parenthesis space is space continous space at space straight x equals 0 comma space limit as straight x rightwards arrow 0 of straight f space left parenthesis straight x right parenthesis space equals limit as straight x rightwards arrow 0 of straight f space left parenthesis 0 right parenthesis Thus space straight R. straight H. straight L space equals space limit as straight x rightwards arrow 0 of straight f space left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 of straight f space left parenthesis 0 plus straight h right parenthesis equals space limit as straight h rightwards arrow 0 of space space fraction numerator square root of 1 plus bh end root minus 1 over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of space fraction numerator square root of 1 plus bh end root minus 1 over denominator straight h end fraction space straight x space fraction numerator square root of 1 plus bh end root plus 1 over denominator square root of 1 plus bh end root plus 1 end fraction equals space limit as straight h rightwards arrow 0 of space fraction numerator 1 plus bh minus 1 over denominator straight h left parenthesis square root of 1 plus bh end root plus 1 right parenthesis end fraction equals space limit as straight h rightwards arrow 0 of space fraction numerator straight b over denominator square root of 1 plus bh end root plus 1 end fraction equals straight b over 2 Given space that space straight f space left parenthesis straight x right parenthesis space equals space 2 rightwards double arrow space limit as straight x rightwards arrow 0 of straight f space left parenthesis straight x right parenthesis space space equals space straight f left parenthesis 0 right parenthesis rightwards double arrow space straight b over 2 space space equals space 2 space rightwards double arrow space straight b equals space 4 Similarly comma space straight L. straight H. straight L space equals space limit as straight x rightwards arrow 0 of straight f space left parenthesis straight x right parenthesis space equals space limit as straight x rightwards arrow 0 of straight f space left parenthesis 0 minus straight h right parenthesis equals space limit as straight h rightwards arrow 0 of fraction numerator sin space left parenthesis straight a plus 1 right parenthesis left parenthesis 0 minus straight h right parenthesis plus 2 sin left parenthesis 0 minus straight h right parenthesis over denominator 0 minus straight h end fraction equals space limit as straight h rightwards arrow 0 of fraction numerator negative sin left parenthesis straight a plus 1 right parenthesis straight h minus 2 space sin space straight h over denominator negative straight h end fraction equals space limit as straight h rightwards arrow 0 of fraction numerator negative sin space left parenthesis straight a plus 1 right parenthesis straight h over denominator negative straight h end fraction space plus space limit as straight h rightwards arrow 0 of fraction numerator negative 2 sin space straight h over denominator negative straight h end fraction equals space limit as straight h rightwards arrow 0 of fraction numerator sin left parenthesis straight a plus 1 right parenthesis straight h over denominator straight h end fraction fraction numerator left parenthesis straight a plus 1 right parenthesis over denominator left parenthesis straight a plus 1 right parenthesis end fraction space plus space 2 stack space lim with straight h rightwards arrow 0 below fraction numerator sin space straight h over denominator straight h end fraction equals space straight a plus 1 plus 2 space space space space space space open square brackets therefore space lim space fraction numerator sin space straight theta over denominator straight theta end fraction equals 1 space close square brackets Given space that space straight f space left parenthesis straight x right parenthesis equals 2 rightwards double arrow stack space lim with straight x space rightwards arrow 0 below space straight f space left parenthesis straight x right parenthesis space equals space straight f space left parenthesis 0 right parenthesis rightwards double arrow space straight a plus 1 plus space 2 space space equals space 2 rightwards double arrow space straight a space equals negative 1

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