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Question

Lt subscript x not stretchy rightwards arrow 1 end subscript space 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

hintHint:

We can apply L'Hopital's rule, also commonly spelled L'Hospital's rule, whenever direct substitution of a limit yields an indeterminate form. This means that the limit of a quotient of functions (i.e., an algebraic fraction) is equal to the limit of their derivatives.
In this question, we have to find value of Lt subscript x not stretchy rightwards arrow 1 end subscript space 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.

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


    Lt subscript x not stretchy rightwards arrow 1 end subscript space 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
    Lt subscript x not stretchy rightwards arrow 1 end subscript space 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 has indeterminate form infinity over infinity, but we can factor and reduce.
    F a c t o r s space o f space 2 x squared plus x minus 3 space equals space left parenthesis 2 x plus 3 right parenthesis left parenthesis x minus 1 right parenthesis space o r space left parenthesis 2 x plus 3 right parenthesis left parenthesis square root of x plus 1 right parenthesis left parenthesis square root of x minus 1 right parenthesis
    Lt subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator left parenthesis 2 x minus 3 right parenthesis left parenthesis square root of x minus 1 right parenthesis over denominator left parenthesis 2 x plus 3 space right parenthesis left parenthesis x minus 1 right parenthesis end fraction space equals space Lt subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator left parenthesis 2 x minus 3 right parenthesis over denominator left parenthesis 2 x plus 3 space right parenthesis left parenthesis square root of x plus 1 right parenthesis end fraction
    On substituting, We get
    fraction numerator left parenthesis 2 minus 3 right parenthesis over denominator left parenthesis 2 plus 3 space right parenthesis left parenthesis square root of 1 plus 1 right parenthesis end fraction space equals space fraction numerator negative 1 over denominator 10 end fraction

    We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means fraction numerator infinity over denominator infinity space end fraction space o r 0 over 0 space.

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