Maths-
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Question

Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction equals

  1. 1
  2. 1 half
  3. 2 over 3
  4. 3 over 2

Hint:

In the question we will use L Hospital theorem, fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals fraction numerator f apostrophe left parenthesis x right parenthesis over denominator g apostrophe left parenthesis x right parenthesis end fraction, then we will put the value of x.

The correct answer is: 2 over 3


    Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction
u sin g space L space h o s p i t a l space t h e o r e m comma space
equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 plus x end root minus begin display style fraction numerator d blank over denominator d x end fraction end style cube root of 1 minus x end root over denominator begin display style fraction numerator d blank over denominator d x end fraction end style x end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator begin display style fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style minus begin display style fraction numerator negative 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction end style over denominator 1 end fraction
equals Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 over denominator 3 open parentheses 1 plus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus x close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction
equals fraction numerator 1 over denominator 3 open parentheses 1 plus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction plus fraction numerator 1 over denominator 3 open parentheses 1 minus 0 close parentheses to the power of begin display style 2 over 3 end style end exponent end fraction
equals 1 third plus 1 third equals 2 over 3

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