Let [y] denote the greatest integer less than or equal to y. If f:(0,∞)→N is defined by f(x)=[x2+x+1x2+1]+[4x2+x+22x2+1]+[9x2+x+33x2+1]+⋯+[n2x2+x+nnx2+1] for n∈N, then the value of limn→∞⎛⎜⎜⎜⎝f(x)−n(f(x))2−n3(n+2)4⎞⎟⎟⎟⎠ is

Let [y] denote the greatest integer less than or equal to y. If f:(0,�

1.	Let [y] denote the greatest integer less than or equal to y. If f:(0,∞)→N is defined by f(x)=[x2+x+1x2+1]+[4x2+x+22x2+1]+[9x2+x+33x2+1]+⋯+[n2x2+x+nnx2+1] for n∈N, then the value of limn→∞⎛⎜⎜⎜⎝f(x)−n(f(x))2−n3(n+2)4⎞⎟⎟⎟⎠ is
Answer» Let $[y]$ denote the greatest integer less than or equal to $y .$ If $f : (0, \infty) \to N$ is defined by $f (x) = [\frac{x^{2} + x + 1}{x^{2} + 1}] + [\frac{4 x^{2} + x + 2}{2 x^{2} + 1}] + [\frac{9 x^{2} + x + 3}{3 x^{2} + 1}] + \dots + [\frac{n^{2} x^{2} + x + n}{n x^{2} + 1}]$ for $n \in N,$ then the value of $lim n \to \infty ⎛ ⎜⎜⎜ ⎝ \frac{f (x) - n}{{(f (x))}^{2} - \frac{n^{3} (n + 2)}{4}} ⎞ ⎟⎟⎟ ⎠$ is