Find the value of lim x -x-2 x-3 x- -n x-n2 where - is an greatest integer function-A- xB- x-2C- 2 xD- 0

The correct option is B We have, lim_x→∞[x]+[2x]+[3x]+....[nx]/n^2 we know greatest integer of [x] lies between x 1 to x x 1 [x] ≤ x ......(1) Similarly 2x 1 ...

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Standard XII

Mathematics

Definition of Sets

Find the valu...

Question

Find the value of $lim x \to \infty \frac{[x] + [2 x] + [3 x] + . . . . [n x]}{n^{2}}$ where [.] is an greatest integer function.

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Solution

The correct option is A

We have, $lim x \to \infty \frac{[x] + [2 x] + [3 x] + . . . . [n x]}{n^{2}}$

we know greatest integer of [x] lies between x-1 to x

$x - 1 < [x] \leq x$ ......(1)

Similarly

$2 x - 1 < [2 x] \leq 2 x$ .....(2)

$3 x - 1 < [3 x] \leq 3 x$ .....(3)

.

.

.

.

$n x - 1 < [n x] \leq n x$

Adding all these terms

We have,

$x - 1 + 2 x - 1 + . . . n x - 2 < [x] + [2 x] + . . . . . [n x] \leq x + 2 x + . . . . n x$

$(x + 2 x + 3 x + . . . n x) - n < [x] + [2 x] + . . . . . [n x] \leq x + 2 x + . . . . n x$

$(1 + 2 + 3 + . . . n) x - n < [x] + [2 x] + [3 x] + . . . . . [n x] \leq (1 + 2 + 3..... N) x$

Sum of n terms $\sum n = \frac{n (n + 1)}{2}$

$\frac{n (n + 1) x}{2} - n$ < [x]+[2x]+[3x]+...[nx] $\leq$ $\frac{n (n + 1)}{2} . x$

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Find the value of limx→∞[x]+[2x]+[3x]+....[nx]n2 where [.] is an greatest integer function.

Find the value of $lim x \to \infty \frac{[x] + [2 x] + [3 x] + . . . . [n x]}{n^{2}}$ where [.] is an greatest integer function.