The value of \( \lim _{n \rightarrow \infty} \frac{1}{n^{3}}\left(\left[1^{2} x+1^{2}\right]+\le...

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The value of \( \lim _{n \rightarrow \infty} \frac{1}{n^{3}}\left(\left[1^{2} x+1^{2}\right]+\left[2^{2} x+2^{2}\right]+\ldots+\left[n^{2} x+n^{2}\right]\right) \) (where \( [\cdot] \) denotes the greatest integer function) is
(a) \( \frac{x}{3} \)
(b) \( x+\frac{1}{3} \)
(c) \( \frac{x}{3}+\frac{1}{3} \)
(d) None of these
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