Home
Class 12
MATHS
If [x] denotes the greatest integer less...

If `[x]` denotes the greatest integer less than or equal to x, then evaluate `lim_(ntooo) (1)/(n^(2))([1.x]+[2.x]+[3.x]+...+[n.x]).`

Text Solution

Verified by Experts

The correct Answer is:
`x//2`
`underset(ntooo)lim(1)/(n^(2))([1.x]+[2.x]+[3.x]+...+[n.x])`
`=underset(ntooo)lim{(sum_(r=1)^(n)[rx])/(n^(2))}`
`=underset(ntooo)lim{(sum_(r=1)^(n)(rx-{rx}))/(n^(2))}`
`=underset(ntooo)lim((x.(n(n+1))/(2))/(n^(2))-sum_(r=1)^(n)({rx})/(n^(2)))`
`=underset(ntooo)lim((x.(1+(1)/(n)))/(2)-sum_(r=1)^(n)({rx})/(n^(2)))`
`=x/2-0=x/2`
Promotional Banner

Topper's Solved these Questions

  • LIMITS

    CENGAGE|Exercise Exercise 2.4|5 Videos

  • LIMITS

    CENGAGE|Exercise Exercise 2.5|12 Videos

  • LIMITS

    CENGAGE|Exercise Exercise 2.2|7 Videos

  • JEE 2019

    CENGAGE|Exercise Chapter 10|9 Videos

  • LINEAR COMBINATION OF VECTORS, DEPENDENT AND INDEPENDENT VECTORS

    CENGAGE|Exercise DPP 1.2|10 Videos

Similar Questions

Explore conceptually related problems

If [x] denotes the greatest integer less than or equal to x, then evaluate lim_(n rarr oo)(1)/(n^(3)){[1^(2)x]+[2^(2)x]+[3^(2)x]+...}[n^(2)x]}

If [x] denotes the greatest integer less than or equal to x, then ["log"_(10) 6730.4]=

If [x] denotes the greatest integer less than or equal to x, then the value of lim_(x rarr1)(1-x+[x-1]+[1-x]) is

If [x] denotes the greatest integer less then or equal to x, then [ (6sqrt(6) + 14)^(2n +1)]

if [x] denotes the greatest integer less than or equal to x, than lim_(xrarr0)(x[x])/(sin|x|) , is

If [x] denotes the greatest integer less than or equal to x then the value of int_(0)^(2)(|x-2|+[x])dx is equal to

Let [x] denote the greatest integer less than or equal to x . Then, int_(1)^(-1)[x]dx=?

If [x] denotes the greatest integer less than or equal to x, then find the value of the integral int_(0)^(2)x^(2)[x]dx

Let [x] denote the greatest integer less than or equal to x . Then, int_(0)^(1.5)[x]dx=?