Limits and Derivatives

Limits and Derivatives

Question

Evaluate  space space space space space limit as straight x rightwards arrow 0 of fraction numerator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x cubed end root space minus square root of 1 plus straight x end root end fraction

Answer

It is of space space space space 0 over 0 space form.
 space space therefore space space space space limit as straight x rightwards arrow 0 of fraction numerator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x cubed end root minus square root of 1 plus straight x end root end fraction equals limit as straight x rightwards arrow 0 of fraction numerator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x squared end root minus square root of 1 plus straight x end root end fraction straight x fraction numerator square root of 1 plus straight x squared end root plus square root of 1 plus straight x end root over denominator square root of 1 plus straight x squared end root plus square root of 1 plus straight z end root end fraction straight x fraction numerator square root of 1 plus straight x cubed end root minus square root of 1 plus straight x end root over denominator square root of 1 plus straight x cubed end root minus square root of 1 plus straight x end root end fraction
   
    space space space space space space space space space equals space limit as straight x rightwards arrow 0 of fraction numerator open square brackets left parenthesis square root of 1 plus straight x squared end root right parenthesis squared minus left parenthesis square root of 1 plus straight x end root right parenthesis squared close square brackets over denominator left square bracket left parenthesis square root of 1 plus straight x cubed end root right parenthesis minus left parenthesis square root of 1 plus straight x end root right parenthesis squared right square bracket end fraction straight x fraction numerator square root of 1 plus straight x cubed end root plus square root of 1 plus straight x end root over denominator square root of 1 plus straight x squared end root plus square root of 1 plus straight x end root end fraction
     space space space space space space space equals space space limit as straight x rightwards arrow 0 of fraction numerator 1 over denominator left parenthesis straight x plus 1 right parenthesis end fraction straight x fraction numerator square root of 1 plus straight x cubed end root plus square root of 1 plus straight x end root over denominator square root of 1 plus straight x squared end root plus square root of 1 plus straight x end root end fraction equals fraction numerator 1 over denominator 0 plus 1 end fraction straight x fraction numerator square root of 1 plus 0 end root plus square root of 1 plus 0 end root over denominator square root of 1 plus 0 end root plus square root of 1 plus 0 end root end fraction
    space space space space space space space space space space space space equals space 1 space cross times space 2 over 2 equals 1

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