Maths-
General
Easy

Question

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis 1 plus x right parenthesis to the power of 1 fifth end exponent minus left parenthesis 1 minus x right parenthesis to the power of 1 half end exponent over denominator x end fraction

  1. 1 fourth
  2. 4
  3. 1
  4. 0

Hint:

In this question we will first use L' Hospital theorem and then simplify the equation and later put the value of x.

The correct answer is: 1 fourth


    space space space space L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis 1 plus x right parenthesis to the power of 1 fifth end exponent minus left parenthesis 1 minus x right parenthesis to the power of 1 half end exponent over denominator x end fraction
U sin g space L apostrophe space H o s p i t a l space t h e o r e m comma space
equals L t subscript x not stretchy rightwards arrow 0 end subscript 1 fifth left parenthesis 1 plus x right parenthesis to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent minus 1 half left parenthesis 1 minus x right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent open parentheses negative 1 close parentheses
equals L t subscript x not stretchy rightwards arrow 0 end subscript 1 fifth left parenthesis 1 plus x right parenthesis to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent plus 1 half left parenthesis 1 minus x right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent
equals 1 fifth cross times open parentheses 1 plus 0 close parentheses to the power of fraction numerator negative 4 over denominator 5 end fraction end exponent plus 1 half left parenthesis 1 minus 0 right parenthesis to the power of fraction numerator negative 1 over denominator 2 end fraction end exponent
equals 1 fifth plus 1 half
equals fraction numerator 2 plus 5 over denominator 10 end fraction
equals 7 over 10.

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