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Question

text  If  end text f left parenthesis 9 right parenthesis equals 9 comma f to the power of apostrophe left parenthesis 9 right parenthesis equals 4 comma text  then  end text space text  Lt end text subscript x not stretchy rightwards arrow 9 end subscript fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction text  equals  end text

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

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 oftext  If  end text f left parenthesis 9 right parenthesis equals 9 comma f to the power of apostrophe left parenthesis 9 right parenthesis equals 4 comma text  then  end text space text  Lt end text subscript x not stretchy rightwards arrow 9 end subscript fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction text  equals  end text.

The correct answer is: 4


    text  If  end text f left parenthesis 9 right parenthesis equals 9 comma f to the power of apostrophe left parenthesis 9 right parenthesis equals 4 comma text  then  end text space text  Lt end text subscript x not stretchy rightwards arrow 9 end subscript fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction text  equals  end text
    We first try substitution:
    space text  Lt end text subscript x not stretchy rightwards arrow 9 end subscript fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction space equals space text  Lt end text subscript x not stretchy rightwards arrow 9 end subscript fraction numerator square root of f left parenthesis 9 right parenthesis end root minus 3 over denominator square root of 9 minus 3 end fraction text  =  end text 0 over 0 text          ( Given f(9)=9 ) end text
    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.
    space text  Lt end text subscript x not stretchy rightwards arrow 9 end subscript fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction text           ( Applying Lh rule ;  end text fraction numerator d over denominator d x end fraction square root of f left parenthesis x right parenthesis end root text  ) = end text fraction numerator 1 over denominator text 2 end text square root of f left parenthesis x right parenthesis end root end fraction f apostrophe left parenthesis x right parenthesis space right parenthesis
space text  Lt end text subscript x not stretchy rightwards arrow 9 end subscript fraction numerator fraction numerator 1 over denominator text 2 end text square root of f left parenthesis x right parenthesis end root end fraction f apostrophe left parenthesis x right parenthesis over denominator fraction numerator 1 over denominator text 2 end text square root of left parenthesis x right parenthesis end root end fraction end fraction text            [ Given  end text f apostrophe left parenthesis x right parenthesis space equals 4 text ]                  end text
Lt subscript x not stretchy rightwards arrow 9 end subscript fraction numerator fraction numerator 4 over denominator text 2 end text square root of 9 end fraction over denominator fraction numerator 1 over denominator text 2 end text square root of 9 end fraction end fraction space equals 4

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

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