Calculus problem #2130

Calculus Level 3

Evaluate \(\displaystyle N = \lim_{x \to \infty} \left(\sqrt{\left \lfloor x^2+x \right \rfloor}-x \right)\).

Details and assumptions

The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).

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