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L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log subscript e begin display style space end style open parentheses 1 plus x cubed close parentheses over denominator sin cubed begin display style space end style x end fraction

  1. log x
  2. 1
  3. x cubed
  4. 0

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 t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log subscript e begin display style space end style open parentheses 1 plus x cubed close parentheses over denominator sin cubed begin display style space end style x end fraction.

The correct answer is: 1


    L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log subscript e begin display style space end style open parentheses 1 plus x cubed close parentheses over denominator sin cubed begin display style space end style x end fraction
    L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log subscript e begin display style space end style open parentheses 1 plus x cubed close parentheses over denominator sin cubed begin display style space end style x end fraction space cross times x cubed over x cubed
    (space W e space k n o w space t h a t space comma space L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log subscript e begin display style space end style open parentheses 1 plus x close parentheses over denominator x end fraction space equals space 1 = 1, L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator sin to the power of blank begin display style space end style x over denominator x end fraction equals space 1 )
    L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log subscript e begin display style space end style open parentheses 1 plus x cubed close parentheses over denominator x cubed end fraction space cross times space L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x cubed over denominator sin cubed begin display style space end style x end fraction 

    On substituting value, We get
    L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator log subscript e begin display style space end style open parentheses 1 plus x cubed close parentheses over denominator x cubed end fraction space cross times space L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x cubed over denominator sin cubed begin display style space end style x end fraction = 1 x 1 = 1

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

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