If \( f:(0, \infty) \rightarrow N \) and \( f(x)=\left[\frac{x^{2}+...

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If \( f:(0, \infty) \rightarrow N \) and \( f(x)=\left[\frac{x^{2}+x+1}{x^{2}+1}\right]+\left[\frac{4 x^{2}+x+2}{2 x^{2}+1}\right]+\left[\frac{9 x^{2}+x+3}{3 x^{2}+1}\right]+\ldots+\left[\frac{n^{2} x^{2}+x+n}{n x^{2}+1}\right], n \in N \) then
P
W find the value of \( \lim _{n \rightarrow \infty}\left[\frac{f(x)-n}{(f(x))^{2}-\frac{n^{3}(n+2)}{4}}\right] \).
[Note \( :[y] \) denotes the greatest integer less than or equal to \( y \).]
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