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Question

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

  1. 27 over 16
  2. 9 over 4
  3. 16 over 27
  4. 4 over 9

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

The correct answer is: 27 over 16


    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction
    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses 1 cubed plus 2 cubed plus midline horizontal ellipsis times plus n cubed close parentheses squared over denominator open parentheses 1 squared plus 2 squared plus midline horizontal ellipsis plus n squared close parentheses cubed end fraction space space left curly bracket space W e space k n o w space t h a t space
sum from n equals 1 to n of space n cubed space equals space fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction comma sum from n equals 1 to n of space n squared space equals space fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n space plus 1 right parenthesis over denominator 6 end fraction space right curly bracket
L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction close parentheses squared over denominator open parentheses fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n space plus 1 right parenthesis over denominator 6 end fraction close parentheses cubed end fraction space equals space L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n open parentheses fraction numerator n to the power of 4 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction close parentheses over denominator open parentheses fraction numerator n cubed left parenthesis n plus 1 right parenthesis cubed left parenthesis 2 n space plus 1 right parenthesis cubed over denominator 216 end fraction close parentheses end fraction space equals space L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n squared open parentheses fraction numerator left parenthesis n plus 1 right parenthesis over denominator 16 end fraction close parentheses over denominator open parentheses left parenthesis 2 n space plus 1 right parenthesis cubed over 216 close parentheses end fraction
L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n squared open parentheses fraction numerator left parenthesis n plus 1 right parenthesis over denominator 16 end fraction close parentheses over denominator open parentheses left parenthesis 2 n space plus 1 right parenthesis cubed over 216 close parentheses end fraction space equals L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n cubed open parentheses fraction numerator space left parenthesis 1 plus begin display style 1 over n end style right parenthesis over denominator 16 end fraction close parentheses over denominator n cubed open parentheses left parenthesis 2 plus begin display style 1 over n end style right parenthesis cubed over 216 close parentheses end fraction
p u t space x equals infinity space a s space w e space k n o w space t h a t 1 over infinity equals 0
s o comma space fraction numerator begin display style 1 over 16 end style over denominator begin display style 8 over 216 end style end fraction space equals 27 over 16

    In general, we say that f(x) tends to a real limit l as x tends to infinity if, however small a distance we choose, f(x) gets closer than that distance to l and stays closer as x increases. f(x) = infinity

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