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L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis comma 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

Hint:

In the question we will first use the formula of sum of cube of n natural numbers then we will use L hospital theorem, fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals fraction numerator f apostrophe left parenthesis x right parenthesis over denominator g apostrophe open parentheses x close parentheses end fraction, twice.

The correct answer is: 1 fourth


    L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus midline horizontal ellipsis comma plus n cubed over denominator n squared open parentheses n squared plus 1 close parentheses end fraction
equals L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style 1 fourth end style n squared open parentheses n plus 1 close parentheses squared over denominator n squared open parentheses n squared plus 1 close parentheses end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n squared plus 2 n plus 1 over denominator n squared plus 1 end fraction
u sin g space L space h o s p i t a l space t h e o r e m comma
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d n squared plus 2 n plus 1 over denominator d n end fraction end style over denominator begin display style fraction numerator d n squared plus 1 over denominator d n end fraction end style end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator 2 n plus 2 over denominator 2 n end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator n plus 1 over denominator n end fraction
A g a i n comma space u sin g space L space h o s p i t a l space t h e o r e m comma
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style fraction numerator d n plus 1 over denominator d n end fraction end style over denominator begin display style fraction numerator d n over denominator d n end fraction end style end fraction
equals 1 fourth L t subscript n not stretchy rightwards arrow straight infinity end subscript 1 over 1
equals 1 fourth

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