Maths-
General
Easy

Question

Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction equals

  1. 4
  2. 2 square root of 2
  3. fraction numerator 1 over denominator square root of 2 end fraction
  4. square root of 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 x over denominator square root of x plus 4 end root minus 2 end fraction.

The correct answer is: 4


    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction
    We first try substitution:
    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction equalsLt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator 0 over denominator square root of 0 plus 4 end root minus 2 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.
    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator x over denominator square root of x plus 4 end root minus 2 end fraction
    Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator begin display style fraction numerator d over denominator d x end fraction end style x over denominator fraction numerator d over denominator d x end fraction square root of x plus 4 end root minus 0 end fraction space space space space space space left parenthesis space A p p l y space L apostrophe H o p i t a l apostrophe s space r u l e right parenthesis
Lt subscript x not stretchy rightwards arrow 0 end subscript space fraction numerator begin display style 1 end style over denominator begin display style fraction numerator 1 over denominator 2 square root of x plus 4 end root end fraction end style end fraction space equals Lt subscript x not stretchy rightwards arrow 0 end subscript space 2 square root of x plus 4 end root space
p u t space x equals 0 space t h e n comma
space 2 square root of x plus 4 end root space equals space 4

    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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