Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 97: 33

$$1$$

\begin{align*} \lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+1}}{x+1}&=\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+1} / \sqrt{x^{2}}}{(x+1) / \sqrt{x^{2}}}\\ &=\lim _{x \rightarrow \infty} \frac{\sqrt{\left(x^{2}+1\right) / x^{2}}}{(x+1) / x}\\ &=\lim _{x \rightarrow \infty} \frac{\sqrt{1+1 / x^{2}}}{(1+1 / x)}\\ &=\frac{\sqrt{1+0}}{(1+0)}\\ &=1 \end{align*}