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Question

Lt subscript x not stretchy rightwards arrow 4 end subscript fraction numerator a to the power of x minus 1 over denominator b to the power of x minus 1 end fraction left parenthesis a greater than 0 comma b greater than 0 comma b not equal to 1 right parenthesis

  1. log subscript 5 superscript 4
  2. log subscript a superscript b
  3. log space a b
  4. log space a minus space log space b

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 x not stretchy rightwards arrow 4 end subscript fraction numerator a to the power of x minus 1 over denominator b to the power of x minus 1 end fraction left parenthesis a greater than 0 comma b greater than 0 comma b not equal to 1 right parenthesis.

The correct answer is: log subscript 5 superscript 4


    Lt subscript x not stretchy rightwards arrow 4 end subscript fraction numerator a to the power of x minus 1 over denominator b to the power of x minus 1 end fraction left parenthesis a greater than 0 comma b greater than 0 comma b not equal to 1 right parenthesis
    Lt subscript x not stretchy rightwards arrow 4 end subscript fraction numerator a to the power of x minus 1 over denominator b to the power of x minus 1 end fraction space cross times space x over x
L t subscript x not stretchy rightwards arrow 4 end subscript fraction numerator a to the power of x minus 1 over denominator x end fraction space cross times subscript x not stretchy rightwards arrow 4 end subscript fraction numerator x over denominator b to the power of x minus 1 end fraction           
left parenthesis space W e space k n o w space t h a t space L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator a to the power of x minus 1 over denominator x end fraction space space equals space 1 space right parenthesis
    so,
    Lt subscript x not stretchy rightwards arrow 4 end subscript fraction numerator a to the power of x minus 1 over denominator b to the power of x minus 1 end fraction space equals space 1

    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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