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Lt subscript h not stretchy rightwards arrow 0 end subscript space fraction numerator f open parentheses 2 h plus 2 plus h squared close parentheses minus f left parenthesis 2 right parenthesis over denominator f open parentheses h minus h squared plus 1 close parentheses minus f left parenthesis 1 right parenthesis end fraction given that f to the power of † left parenthesis 2 right parenthesis equals 6 text  and  end text f to the power of † left parenthesis 1 right parenthesis equals 4

  1. is equal to 3

  2. does not exist

  3. is equal of 3 over 2

  4. is equal to fraction numerator negative 3 over denominator 2 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 h not stretchy rightwards arrow 0 end subscript space fraction numerator f open parentheses 2 h plus 2 plus h squared close parentheses minus f left parenthesis 2 right parenthesis over denominator f open parentheses h minus h squared plus 1 close parentheses minus f left parenthesis 1 right parenthesis end fraction.

The correct answer is:

is equal to 3


    Lt subscript h not stretchy rightwards arrow 0 end subscript space fraction numerator f open parentheses 2 h plus 2 plus h squared close parentheses minus f left parenthesis 2 right parenthesis over denominator f open parentheses h minus h squared plus 1 close parentheses minus f left parenthesis 1 right parenthesis end fraction given that f to the power of † left parenthesis 2 right parenthesis equals 6 text  and  end text f to the power of † left parenthesis 1 right parenthesis equals 4
    Lt subscript h not stretchy rightwards arrow 0 end subscript space fraction numerator f open parentheses 2 h plus 2 plus h squared close parentheses minus f left parenthesis 2 right parenthesis over denominator f open parentheses h minus h squared plus 1 close parentheses minus f left parenthesis 1 right parenthesis end fraction
    We first try substitution:
    Lt subscript h not stretchy rightwards arrow 0 end subscript space fraction numerator f open parentheses 2 h plus 2 plus h squared close parentheses minus f left parenthesis 2 right parenthesis over denominator f open parentheses h minus h squared plus 1 close parentheses minus f left parenthesis 1 right parenthesis end fraction space equals space fraction numerator f open parentheses 2 close parentheses minus f left parenthesis 2 right parenthesis over denominator f open parentheses 1 close parentheses minus f left parenthesis 1 right parenthesis end fraction space equals 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 h not stretchy rightwards arrow 0 end subscript space fraction numerator f open parentheses 2 h plus 2 plus h squared close parentheses minus f left parenthesis 2 right parenthesis over denominator f open parentheses h minus h squared plus 1 close parentheses minus f left parenthesis 1 right parenthesis end fraction space equals space Lt subscript h not stretchy rightwards arrow 0 end subscript space fraction numerator begin display style fraction numerator d over denominator d x end fraction end style f open parentheses 2 h plus 2 plus h squared close parentheses over denominator begin display style fraction numerator d over denominator d x end fraction end style f open parentheses h minus h squared plus 1 close parentheses end fraction space equals Lt subscript h not stretchy rightwards arrow 0 end subscript space fraction numerator begin display style f apostrophe open parentheses 2 plus 2 h close parentheses end style over denominator begin display style f apostrophe open parentheses 1 minus 2 h close parentheses end style end fraction space space space space space left curly bracket space a p p l y L apostrophe H o p i t a l apostrophe s space r u l e space right curly bracket space
Lt subscript h not stretchy rightwards arrow 0 end subscript space fraction numerator begin display style f open parentheses 2 plus 2 h close parentheses end style over denominator begin display style f open parentheses 1 minus 2 h to the power of blank close parentheses end style end fraction space equals fraction numerator begin display style f apostrophe open parentheses 2 close parentheses end style over denominator begin display style f apostrophe open parentheses 1 close parentheses end style end fraction space equals 6 over 4 equals 3 over 2 

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

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