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Question

The value of limn1n3([12x]+[22x]++[n2x]),xR is equal to (Here [ .] denotes the greatest integer function)

A
x6
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B
x3
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C
x2
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D
0
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Solution

The correct option is B x3
We know k1<[k]k, kR
k2x1<[k2x]k2x

nk=1(k2x1)<nk=1[k2x]nk=1k2x

n(n+1)(2n+1)6xn<nk=1[k2x]n(n+1)(2n+1)6x

n(n+1)(2n+1)x6n6n3<1n3nk=1[k2x]n(n+1)(2n+1)6n3x

Since limnn(n+1)(2n+1)x6n6n3=limnn(n+1)(2n+1)x6n3=x3
By Sandwich theorem of limits,
limn1n3nk=1[k2x]=x3

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