f [x] denotes the greatest integer less than or equal to x ,then limn→∞ [x]+[2x]+[3x]+….+[nx]n2 equal 

f [x] denotes the greatest integer less than or equal to x ,then 

limn[x]+[2x]+[3x]+.+[nx]n2 equal 

  1. A

    x/2

  2. B

    x/3

  3. C

    x

  4. D

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    Solution:

    For any integer k, we have

    kx1<[kx]kxk=1n(kx1)<k=1n[kx]k=1nkx1n2k=1n(kx1)<1n2k=1n[kx]1n2k=1nkxxn2k=1nk1n2<1n2k=1n[kx]xn2k=1nkx21+1n1n2<1n2k=1n[kx]x21+1n

    Now, 

    limnx21+1n1n2=x2 and , limnx21+1n=x2

    =limn1n2n[kx]=x2  [Using Sandwich Theorem]

    i .e.limn[x]+[2x]+[3x]++[nx]n2=x2

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