Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 99: 86

$$1$$

\begin{aligned} \lim _{x \rightarrow \infty} \sqrt{x^{2}+x}-\sqrt{x^{2}-x} &=\lim _{x \rightarrow \infty}[\sqrt{x^{2}+x}-\sqrt{x^{2}-x}] \cdot\left[\frac{\sqrt{x^{2}+x}+\sqrt{x^{2}-x}}{\sqrt{x^{2}+x+\sqrt{x^{2}-x}}}\right]\\ &=\lim _{x \rightarrow \infty} \frac{\left(x^{2}+x\right)-\left(x^{2}-x\right)}{\sqrt{x^{2}+x+\sqrt{x^{2}-x}}}\\ &=\lim _{x \rightarrow \infty} \frac{2 x}{\sqrt{x^{2}+x+\sqrt{x^{2}-x}}} \\ &=\lim _{x \rightarrow \infty} \frac{2}{\sqrt{1+\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}\\ &=\frac{2}{1+1}\\ &=1 \end{aligned}