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Question

Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction equals

  1. 1 over 10
  2. fraction numerator negative 1 over denominator 10 end fraction
  3. 1 fifth
  4. fraction numerator negative 1 over denominator 5 end fraction

Hint:

In this question we will use L Hospital theorem open parentheses i. e. comma space 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 close parentheses and then put the value of x.

The correct answer is: fraction numerator negative 1 over denominator 10 end fraction


    Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator 2 x squared plus x minus 3 end fraction
equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator 2 x squared plus x minus 3 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 1 end subscript fraction numerator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x to the power of begin display style 3 over 2 end style end exponent minus 3 to the power of begin display style 1 half end style end exponent minus 2 x plus 3 over denominator begin display style fraction numerator d blank over denominator d x end fraction end style 2 x squared plus x minus 3 end fraction
equals Lt subscript x not stretchy rightwards arrow 1 end subscript fraction numerator 3 square root of x minus begin display style fraction numerator 3 over denominator 2 square root of x end fraction end style minus 2 over denominator 4 x plus 1 end fraction
equals fraction numerator 3 cross times 1 space minus begin display style 3 over 2 end style cross times 1 minus 2 over denominator 4 cross times 1 space plus 1 end fraction
equals fraction numerator 3 minus begin display style 3 over 2 end style minus 2 over denominator 5 end fraction
equals fraction numerator begin display style fraction numerator 6 minus 3 minus 4 over denominator 2 end fraction end style over denominator 5 end fraction
equals fraction numerator negative 1 over denominator 10 end fraction

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