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Question

L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator open parentheses 1 minus e to the power of x close parentheses sin begin display style space end style x over denominator x squared plus x cubed end fraction equals

  1. 1
  2. 0
  3. -1
  4. e squared

hintHint:

In this question, we have to find value of L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator open parentheses 1 minus e to the power of x close parentheses sin begin display style space end style x over denominator x squared plus x cubed end fraction.

The correct answer is: -1


    L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator open parentheses 1 minus e to the power of x close parentheses sin begin display style space end style x over denominator x squared plus x cubed end fraction
    L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator open parentheses 1 minus e to the power of x close parentheses sin begin display style space end style x over denominator x squared plus x cubed end fraction space equals space fraction numerator open parentheses 1 minus e to the power of 0 close parentheses sin begin display style space end style 0 over denominator 0 end fraction space equals space 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.
    L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator stretchy left parenthesis 1 minus straight e to the power of straight x stretchy right parenthesis sin begin display style space end style straight x over denominator straight x left parenthesis 1 plus straight x right parenthesis straight x end fraction space space space left parenthesis W e space k n o w space t h a t space space L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator sin begin display style space end style straight x over denominator straight x end fraction equals 1 comma space L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator left parenthesis straight e to the power of straight x minus 1 right parenthesis over denominator straight x end fraction equals 1 right parenthesis
L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator negative stretchy left parenthesis straight e to the power of straight x minus 1 stretchy right parenthesis over denominator straight x left parenthesis 1 plus straight x right parenthesis end fraction cross times fraction numerator sinx space over denominator straight x end fraction
L t subscript straight x not stretchy rightwards arrow 0 end subscript fraction numerator negative 1 over denominator left parenthesis 1 plus straight x right parenthesis end fraction
p u t space x equals 0 space t h e n comma
fraction numerator negative 1 over denominator left parenthesis 1 plus 0 right parenthesis end fraction equals space minus 1

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