Tính giới hạn sau: lim n → + ∞ ( √ n^2 + n − √ n^2 + 1 ) .

\[\mathop {\lim }\limits_{n \to + \infty } \left( {\sqrt {{n^2} + n} - \sqrt {{n^2} + 1} } \right) = \mathop {\lim }\limits_{n \to + \infty } \frac{{{n^2} + n - {n^2} - 1}}{{\sqrt {{n^2} + n} + \sqrt {{n^2} + 1} }} = \mathop {\lim }\limits_{n \to + \infty } \frac{{n - 1}}{{\sqrt {{n^2} + n} + \sqrt {{n^2} + 1} }}\]

\[ = \mathop {\lim }\limits_{n \to + \infty } \frac{{1 - \frac{1}{n}}}{{\sqrt {1 + \frac{1}{n}} + \sqrt {1 + \frac{1}{{{n^2}}}} }} = \frac{1}{{1 + 1}} = \frac{1}{2}.\]