詳解： \(\lim\limits_{x \to \infty } (\sqrt {{x^2} + x}  - x) = \lim\limits_{x \to \infty } \frac{{\sqrt {{x^2} + x}  - x}}{1}\)

\(\lim\limits_{x \to \infty } \frac{{\left( {\sqrt {{x^2} + x}  - x} \right)\left( {\sqrt {{x^2} + x}  + x} \right)}}{{\left( {\sqrt {{x^2} + x}  + x} \right)}}\) \(\lim\limits_{x \to \infty } \frac{{{x^2} + x - {x^2}}}{{\sqrt {{x^2} + x}  + x}}\)\(\lim\limits_{x \to \infty } \frac{x}{{\sqrt {{x^2} + x}  + x}}\)\(\lim\limits_{x \to \infty } \frac{x}{{x + x}} = \frac{1}{2}\)