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Question

Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction

  1. 1 half
  2. 1
  3. 2
  4. 0

hintHint:

We can apply L'Hopital's rule, also commonly spelled L'Hospital's rule, whenever direct substitution of a limit yields an indeterminate form. This means that the limit of a quotient of functions (i.e., an algebraic fraction) is equal to the limit of their derivatives.
In this question, we have to find value of Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction.

The correct answer is: 1 half


    Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction
    We first try substitution:
    Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction = Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus 0 plus 0 close parentheses end root minus 1 over denominator 0 end fraction space equals space 0 over 0
    Since the limit is in the form 0 over 0, it is indeterminate—we don’t yet know what is it. We need to do some work to put it in a form where we can determine the limit.
    Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator square root of open parentheses 1 plus x plus x squared close parentheses end root minus 1 over denominator x end fraction = Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 left parenthesis 1 plus 2 x right parenthesis over denominator 2 square root of open parentheses 1 plus x plus x squared close parentheses end root end fraction   ( L'Hopital's Rule for zero over zero ; Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction space equals space fraction numerator f apostrophe left parenthesis x right parenthesis over denominator g apostrophe left parenthesis x right parenthesis end fraction  )
    On substituting, We get
    Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 left parenthesis 1 plus 0 right parenthesis over denominator 2 square root of open parentheses 1 plus 0 plus 0 close parentheses end root end fraction space equals Lt subscript x not stretchy rightwards arrow 0 end subscript 1 half space equals space 1 half

    We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 space o r space fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

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