Maths-
General
Easy

Question

Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction equals

  1. 1
  2. 1 half
  3. 2 over 3
  4. 3 over 2

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 0 end subscript space fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction.

The correct answer is: 2 over 3


    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction
    We first try substitution:
    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction equals Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator cube root of 1 plus 0 end root minus cube root of 1 minus 0 end root over denominator 0 end fraction space equals 0 over 0 space
S i n c e space t h e space l i m i t space i s space i n space t h e space f o r m space space 0 space o v e r space 0 space comma space i t space i s space i n d e t e r m i n a t e space w e space d o n ’ t space y e t space k n o w space w h a t space i s space i t. space W e space n e e d space t o space d o space s o m e space w o r k space t o space p u t space i t space i n space a space f o r m space
w h e r e space w e space c a n space d e t e r m i n e space t h e space l i m i t.
Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator cube root of 1 plus x end root minus cube root of 1 minus x end root over denominator x end fraction space equals Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 1 over denominator 3 left parenthesis 1 plus x right parenthesis begin display style bevelled fraction numerator 2 over denominator 3 space end fraction end style cross times space left parenthesis 1 minus x right parenthesis bevelled 2 over 3 end fraction space left parenthesis u sin g space L apostrophe H o p i t a l apostrophe s space r u l e right parenthesis
p u t space x equals 0
Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 1 over denominator 3 left parenthesis 1 plus x right parenthesis to the power of begin display style bevelled 2 over 3 end style end exponent cross times space left parenthesis 1 minus x right parenthesis to the power of begin display style bevelled 2 over 3 end style to the power of blank end exponent end fraction space equals space 1 third

    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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