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Continuity And Differentiability

Question
CBSEENMA12036015
Wired Faculty App

limit as straight n rightwards arrow infinity of space open parentheses fraction numerator left parenthesis straight n plus 1 right parenthesis left parenthesis straight n plus 2 right parenthesis....3 straight n over denominator straight n to the power of 2 straight n end exponent end fraction close parentheses to the power of 1 divided by straight n end exponent is equal to

  • 18/e4

  • 27/e2

  • 9/e2

  • 3 log 3-2

Solution

B.

27/e2

Let space straight I space equals space limit as straight n stack rightwards arrow infinity with space on top of open parentheses fraction numerator left parenthesis straight n plus 1 right parenthesis. left parenthesis straight n plus 2 right parenthesis..... left parenthesis 3 straight n right parenthesis over denominator straight n to the power of 2 straight n end exponent end fraction close parentheses to the power of 1 over straight n end exponent space equals space space limit as straight n stack rightwards arrow infinity with space on top of open parentheses fraction numerator left parenthesis straight n plus 1 right parenthesis. left parenthesis straight n plus 2 right parenthesis..... left parenthesis straight n plus 2 straight n right parenthesis over denominator straight n to the power of 2 straight n end exponent end fraction close parentheses to the power of 1 over straight n end exponent equals space limit as straight n stack rightwards arrow infinity with space on top of open square brackets open parentheses fraction numerator straight n plus 1 over denominator straight n end fraction close parentheses plus open parentheses fraction numerator straight n plus 2 over denominator straight n end fraction close parentheses.... open parentheses fraction numerator straight n plus 2 straight n over denominator straight n end fraction close parentheses close square brackets to the power of 1 over straight n end exponent
taking space log space on space both space sides space comma we space get log space straight I space equals space limit as straight n stack rightwards arrow infinity with space on top of open square brackets log space open curly brackets open parentheses 1 plus 1 over straight n close parentheses open parentheses 1 plus 2 over straight n close parentheses.... open parentheses 1 plus fraction numerator 2 straight n over denominator straight n end fraction close parentheses close curly brackets close square brackets rightwards double arrow space log space straight I space equals space limit as straight n stack rightwards arrow infinity with space on top of 1 over straight n open square brackets log open parentheses 1 plus 1 over straight n close parentheses plus space log space open parentheses 1 plus 2 over straight n close parentheses space plus..... plus space log space open parentheses 1 plus fraction numerator 2 straight n over denominator straight n end fraction close parentheses close square brackets rightwards double arrow space log space straight I space equals limit as straight n stack rightwards arrow infinity with space on top of 1 over straight n sum from straight r space equals 1 to 2 straight n of space log space open parentheses 1 plus straight r over straight n close parentheses
rightwards double arrow space log space straight I space equals space integral subscript 0 superscript 2 space log space left parenthesis 1 plus straight x right parenthesis space dx rightwards double arrow space log space straight I space equals space open square brackets log space space left parenthesis 1 space plus straight x right parenthesis. straight x space minus space integral fraction numerator 1 over denominator 1 plus straight x end fraction. straight x space dx close square brackets subscript 0 superscript 2 rightwards double arrow space log space space straight I space equals space left square bracket log space left parenthesis 1 space plus straight x right parenthesis. straight x right square bracket subscript 0 superscript 2 space minus integral subscript 0 superscript 2 open parentheses 1 minus fraction numerator 1 over denominator 1 plus straight x end fraction close parentheses dx rightwards double arrow space log space space straight I space equals space 2. space log space 3 space minus space left square bracket straight x minus log space vertical line 1 plus straight x vertical line subscript 0 superscript 2 rightwards double arrow space log space space straight I space equals space 3. space log space 3 space minus 2 rightwards double arrow space log space space straight I space equals space space log space 27 space minus 2 therefore comma space straight l space equals space straight e to the power of log space 27 minus 2 end exponent space equals space 27. straight e to the power of negative 2 end exponent space equals space 27 over straight e squared

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