KVPY SX SB Exam 2018 Question Paper

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Question : 86 of 120

Marks: +1, -0

Suppose the limit

L= \lim↙{n →∞}√n∫↖{1}↙{0} 1/ (1 +x^2 ) ^n dx

, Exists and is larger than

1/2

. Then

$1/ 2$ < L < 2
2 < L < 3
3 < L < 4
L ≥ 4

Solution:

(1 + x^2 ) ^n > 1 + nx^2

1/(1 + x^2 ) ^n<1/{1 +n x^2 }

∫↖{1}↙{0} 1/ (1 +x^2 ) ^n <∫↖{1}↙{0} 1/ {1 +nx^2

=1/√n\tan^{-1}√nx|_0^1=1/√n \tan^{-1}√n

∴

\lim↙{n →∞}√n∫↖{1}↙{0} 1/ (1 +x^2 ) ^n dx

<\lim↙{n →∞}√n1/√n\tan^{-1}√n

=π\/2<2

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