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Lt subscript n not stretchy rightwards arrow straight infinity end subscript space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis times plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction

  1. 1
  2. 4
  3. 1 fourth
  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 Lt subscript n not stretchy rightwards arrow straight infinity end subscript space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis times plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction.

The correct answer is: 1 fourth


    Lt subscript n not stretchy rightwards arrow straight infinity end subscript space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis times plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction
    We first try substitution:
    Lt subscript n not stretchy rightwards arrow straight infinity end subscript space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis times plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction = infinity over infinity
    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 n not stretchy rightwards arrow straight infinity end subscript space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis times plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction    (W e space k n o w space t h a t comma space 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis times plus n cubed space space equals space fraction numerator n squared space left parenthesis n space plus space 1 right parenthesis over denominator 4 end fraction )
    Lt subscript n not stretchy rightwards arrow straight infinity end subscript space fraction numerator fraction numerator n squared space left parenthesis n space plus space 1 right parenthesis over denominator 4 end fraction over denominator n squared open parentheses n squared plus 1 close parentheses end fraction space equals space Lt subscript n not stretchy rightwards arrow straight infinity end subscript space 1 fourth
    On substituting, We get
    Lt subscript n not stretchy rightwards arrow straight infinity end subscript space 1 fourth space equals space 1 fourth

    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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