The value of

MATHEMATICS · Single correct · AoPS · Mathematics · Special Functions · Special Functions

Problem

The value of

\lim_{n \to \infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)

, where

\lfloor\cdot\rfloor

denotes the greatest integer function, is (1) 0 (2) 1/2 (3) 2/3 (4) 1/4