Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 98: 80

$$0$$

\begin{aligned} \lim _{x \rightarrow \infty}(\sqrt{x+9}-\sqrt{x+4}) &=\lim _{x \rightarrow \infty}[\sqrt{x+9}-\sqrt{x+4}] \cdot\left[\frac{\sqrt{x+9}+\sqrt{x+4}}{\sqrt{x+9}+\sqrt{x+4}}\right]\\ &=\lim _{x \rightarrow \infty} \frac{(x+9)-(x+4)}{\sqrt{x+9}+\sqrt{x+4}} \\ &=\lim _{x \rightarrow \infty} \frac{5}{\sqrt{x+9}+\sqrt{x+4}}\\ &=\lim _{x \rightarrow \infty} \frac{\frac{\sqrt{x}}{\sqrt{x}}}{\sqrt{1+\frac{9}{x}}+\sqrt{1+\frac{4}{x}}}\\ &=\frac{0}{1+1}\\ &=0 \end{aligned}