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If G left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root then Lim subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis over denominator x minus 1 end fraction has the value

  1. 1 over 24
  2. 1 fifth
  3. negative square root of 24
  4. none of these

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 If G left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root then Lim subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis over denominator x minus 1 end fraction .

The correct answer is:

none of these


    If G left parenthesis x right parenthesis equals negative square root of 25 minus x squared end root then Lim subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis over denominator x minus 1 end fraction
    We first try substitution:
    Lim subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator negative square root of 25 minus x squared end root plus square root of 24 over denominator x minus 1 end fraction space equals fraction numerator negative square root of 25 minus 1 squared end root plus square root of 24 over denominator 1 minus 1 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.
    Lim subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator negative square root of 25 minus x squared end root plus square root of 24 over denominator x minus 1 end fraction space space space space space left square bracket a p p l y space L apostrophe H o p i t a l apostrophe s space r u l e space semicolon fraction numerator d over denominator d x end fraction square root of a minus x to the power of 2 space end exponent end root space equals fraction numerator negative 2 x over denominator 2 square root of a minus x to the power of 2 space end exponent end root end fraction space right square bracket
Lim subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator fraction numerator 2 x over denominator 2 square root of 25 minus x to the power of 2 space end exponent end root end fraction over denominator 1 end fraction space equals Lim subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator x over denominator square root of 25 minus x to the power of 2 space end exponent end root end fraction space
Lim subscript x not stretchy rightwards arrow 1 end subscript space fraction numerator x over denominator square root of 25 minus x to the power of 2 space end exponent end root end fraction equals space fraction numerator 1 over denominator square root of 25 minus 1 to the power of 2 space end exponent end root end fraction space equals 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 or fraction numerator plus-or-minus infinity over denominator plus-or-minus infinity end fraction.

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