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text  If  end text a greater than 0 text  and end text L subscript x not stretchy rightwards arrow a end subscript fraction numerator a to the power of a minus x to the power of a over denominator x to the power of a minus a to the power of a end fraction equals negative 1 text , then  end text bold a equals

  1. 0
  2. 1
  3. -1
  4. straight infinity

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 L subscript x not stretchy rightwards arrow a end subscript fraction numerator a to the power of a minus x to the power of a over denominator x to the power of a minus a to the power of a end fraction equals negative 1 text , then  end text bold a equals ?.

The correct answer is: 1


    text  If  end text a greater than 0 text  and  end text L subscript x not stretchy rightwards arrow a end subscript fraction numerator a to the power of a minus x to the power of a over denominator x to the power of a minus a to the power of a end fraction equals negative 1 text , then  end text bold a equals ?
L subscript x not stretchy rightwards arrow a end subscript fraction numerator a to the power of a minus x to the power of a over denominator x to the power of a minus a to the power of a end fraction equals negative 1
w e space c a n space a l s o space w r i t e comma
L subscript x not stretchy rightwards arrow a end subscript space minus left parenthesis fraction numerator a to the power of a minus x to the power of a over denominator a to the power of a minus x to the power of a end fraction right parenthesis equals negative 1
a element of R to the power of plus space end exponent comma space I n space t h e space g i v e n space o p t i o n space space a equals 1 space i s space s a t i s f i e d space i t.

    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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