Expanding Sequence

$n^2 + \sqrt{k} = n^2 + o(n^2)$ and $n^2 + \sqrt{an-b} = n^2 + o(n^2)$ when $n\to+\infty$ $\forall k, a, b \in \mathbb{N}$ . Therefore: $\lim_{n\to+\infty} \left( \frac{1}{n^2+\sqrt{1}} + \frac{2}{n^2+\sqrt{2}} + \frac{3}{n^2+\sqrt{3}} + ... + \frac{66n-1}{n^2+\sqrt{66n-1}} + \frac{66n}{n^2+\sqrt{66n}} \right) =$ $= \lim_{n\to+\infty} \left( \frac{1}{n^2+o(n^2)} + \frac{2}{n^2+o(n^2)} + \frac{3}{n^2+o(n^2)} + ... + \frac{66n-1}{n^2+o(n^2)} + \frac{66n}{n^2+o(n^2)} \right) =$ $= \lim_{n\to+\infty} \frac{1+2+3+...+66n}{n^2+o(n^2)} =$ $= \lim_{n\to+\infty} \frac{\frac{66n(66n+1)}{2}}{n^2+o(n^2)} =$ $= \lim_{n\to+\infty} \frac{66^2 n^2 + 66n}{2n^2+o(n^2)} =$ $= \lim_{n\to+\infty} \frac{66^2 n^2 + o(n^2)}{2n^2+o(n^2)} =$ $= \lim_{n\to+\infty} \frac{66^2 n^2}{2n^2} = \frac{66^2}{2} = \boxed{2178}$