Solution: Note that \begin{align*} \lim_{n \to \infty} \left( \sqrt{n^2 + n} - n \right) & = \lim_{n \to \infty} \left[ \left( \sqrt{n^2 + n} - n \right) \times \frac{\sqrt{n^2 + n} + n}{\sqrt{n^2 + n} + n} \right] \\ & = \lim_{n \to \infty} \left[ \frac{n^2 + n - n^2}{\sqrt{n^2 +n} + n} \right] \\ & = \lim_{n \to \infty} \left[ \frac{n}{\sqrt{n^2 + n} + n } \right] \\ & = \lim_{n \to \infty} \left[ \frac{\frac{n}{n}}{\sqrt{\frac{n^2 + n}{n^2}} + \frac{n}{n}} \right] \\ & = \lim_{n \to \infty} \left[ \frac{1}{\sqrt{1 + \frac{1}{n}} + 1} \right] \\ & = \frac{1}{2}. \end{align*}