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L t 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

Hint:

In the question first we will use the L hospital theorem, fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals fraction numerator f apostrophe open parentheses x close parentheses over denominator g apostrophe open parentheses x close parentheses end fraction. Then we will put the value of x.

The correct answer is: 1 half


    L t 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
    using L hospital theorem,
    equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d square root of 1 plus x plus x squared end root over denominator d x end fraction end style minus begin display style fraction numerator d blank over denominator d x end fraction end style 1 over denominator begin display style fraction numerator d x over denominator d x end fraction end style end fraction
equals limit as x rightwards arrow 0 of fraction numerator 1 plus 2 x over denominator 2 square root of 1 plus x plus x squared end root end fraction
equals fraction numerator 1 plus 2 cross times 0 over denominator 2 square root of 1 plus 0 plus 0 end root end fraction
equals 1 half

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