Maths-
General
Easy

Question

Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction equals

  1. 1 over 24
  2. 1 fourth
  3. 3 over 2
  4. 1 half

Hint:

In the question we will first put y=square root of 1 plus x end root and convert the x limit to y limit then we will simplify the equation to get the answer.

The correct answer is: 1 over 24


    Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction
L e t space y equals square root of 1 plus x end root space space space space space space space space space space space space space space space space space space space space x rightwards arrow 8
rightwards double arrow y squared equals 1 plus x space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus x end root
rightwards double arrow x equals y squared minus 1 space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow square root of 1 plus 8 end root
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space y rightwards arrow 3
Lt subscript x not stretchy rightwards arrow 8 end subscript space fraction numerator square root of 1 plus square root of 1 plus x end root end root minus 2 over denominator x minus 8 end fraction
equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 1 minus 8 end fraction
equals limit as y rightwards arrow 3 of fraction numerator square root of 1 plus y end root minus 2 over denominator y squared minus 9 end fraction
equals limit as y rightwards arrow 3 of fraction numerator open parentheses square root of 1 plus y end root minus 2 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction
equals limit as y rightwards arrow 3 of fraction numerator 1 plus y minus 4 over denominator open parentheses y squared minus 9 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction
equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator open parentheses y squared minus 3 squared close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction
equals limit as y rightwards arrow 3 of fraction numerator y minus 3 over denominator left parenthesis y minus 3 right parenthesis left parenthesis y plus 3 right parenthesis open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction
equals limit as y rightwards arrow 3 of fraction numerator 1 over denominator open parentheses y plus 3 close parentheses open parentheses square root of 1 plus y end root plus 2 close parentheses end fraction
equals fraction numerator 1 over denominator left parenthesis 3 plus 3 right parenthesis left parenthesis square root of 1 plus 3 end root plus 2 right parenthesis end fraction
equals fraction numerator 1 over denominator 6 cross times 4 end fraction equals 1 over 24

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