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Question

Lt subscript x not stretchy rightwards arrow 3 end subscript space fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction

  1. 7 over 3
  2. fraction numerator negative 7 over denominator 3 end fraction
  3. 1 third
  4. fraction numerator negative 1 over denominator 3 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 3 end subscript space fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction.

The correct answer is: fraction numerator negative 7 over denominator 3 end fraction


    Lt subscript x not stretchy rightwards arrow 3 end subscript space fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction
    We first try substitution:
    Lt subscript x not stretchy rightwards arrow 3 end subscript space fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction = Lt subscript x not stretchy rightwards arrow 3 end subscript space fraction numerator 3 cubed minus 8 cross times 3 squared plus 45 over denominator 2 cross times 3 squared minus 3 cross times 3 minus 9 end fraction space equals space 0 over 0
    Since the limit is in the form 0 over 0, it is indeterminate—we don’t yet know what is it. We need to do some work to put it in a form where we can determine the limit.
    Lt subscript x not stretchy rightwards arrow 3 end subscript space fraction numerator x cubed minus 8 x squared plus 45 over denominator 2 x squared minus 3 x minus 9 end fraction = Lt subscript x not stretchy rightwards arrow 3 end subscript space fraction numerator 3 x squared minus 16 x to the power of blank over denominator 4 x to the power of blank minus 3 end fraction             ( L'Hopital's Rule for zero over zero ; Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator f left parenthesis x right parenthesis to the power of blank over denominator g left parenthesis x right parenthesis end fraction equals fraction numerator f apostrophe left parenthesis 0 right parenthesis to the power of blank over denominator g apostrophe left parenthesis 0 right parenthesis end fraction )
    Lt subscript x not stretchy rightwards arrow 3 end subscript space fraction numerator 3 x squared minus 16 x to the power of blank over denominator 4 x to the power of blank minus 3 end fraction
    On substituting, We get
    Lt subscript x not stretchy rightwards arrow 3 end subscript space fraction numerator 3 cross times 9 minus 16 cross times 3 over denominator 4 cross times 3 minus 3 end fraction space equals space fraction numerator negative 21 over denominator 9 end fraction space equals space fraction numerator negative 7 over denominator 3 end fraction

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

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