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General
Easy

Question

text  If  end text f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root comma text  then  end text L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction equals

  1. 1 over 24
  2. 1 fifth
  3. negative square root of 24
  4. fraction numerator 1 over denominator square root of 24 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 text  If  end text f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root comma text  then  end text L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction.

The correct answer is: fraction numerator 1 over denominator square root of 24 end fraction


    text  If  end text f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root comma text  then  end text L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction
L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis over denominator x minus 1 end fraction equals ?
G i v e n space t h a t comma space space f left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root
i f space x equals 1 space t h e n space space f left parenthesis 1 right parenthesis equals negative square root of 25 minus 1 squared end root
space space space space space space space space space space space space space space space space space space space space space space space space f left parenthesis 1 right parenthesis equals negative square root of 24
    L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator negative square root of 25 minus x squared end root plus space square root of 24 over denominator x minus 1 end fraction space left parenthesis B y space u sin g space L h space r u l e right parenthesis
L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator begin display style fraction numerator d over denominator d x end fraction end style left parenthesis negative square root of 25 minus x squared right parenthesis end root plus space 0 over denominator fraction numerator d over denominator d x end fraction x minus 0 end fraction
L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator begin display style fraction numerator negative 1 over denominator 2 end fraction end style left parenthesis 25 minus x squared right parenthesis to the power of begin display style fraction numerator negative 1 over denominator 2 end fraction end style end exponent left parenthesis negative 2 x right parenthesis over denominator 1 end fraction
L t subscript x not stretchy rightwards arrow 1 end subscript fraction numerator x over denominator square root of 25 minus x squared end root end fraction
fraction numerator 1 over denominator square root of 24 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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