Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 98: 81

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\begin{aligned} \lim _{x \rightarrow \infty}(\sqrt{x^{2}+25}-\sqrt{x^{2}-1}) &=\lim _{x \rightarrow \infty}[\sqrt{x^{2}+25}-\sqrt{x^{2}-1}] \cdot\left[\frac{\sqrt{x^{2}+25}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+25}+\sqrt{x^{2}-1}}\right]\\ &=\lim _{x \rightarrow \infty} \frac{\left(x^{2}+25\right)-\left(x^{2}-1\right)}{\sqrt{x^{2}+25}+\sqrt{x^{2}-1}} \\ &=\lim _{x \rightarrow \infty} \frac{26}{\sqrt{x^{2}+25}+\sqrt{x^{2}-1}}\\ &=\lim _{x \rightarrow \infty} \frac{\frac{26}{x}}{\sqrt{1+\frac{25}{x^{2}}}+\sqrt{\frac{1}{x^{2}}}}\\ &=\frac{0}{1+1}\\ &=0 \end{aligned}