Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 97: 11

$$-1$$

Since \begin{align*} \lim _{t \rightarrow-\infty} \frac{2-t+\sin t}{t+\cos t}&=\lim _{t \rightarrow-\infty} \frac{\frac{2}{t}-1+\frac{\sin t}{t}}{1+\frac{\cos t}{t}}\\ &=\frac{\lim _{t \rightarrow-\infty}\frac{2}{t}-\lim _{t \rightarrow-\infty}(1)+\lim _{t \rightarrow-\infty}\frac{\sin t}{t}}{\lim _{t \rightarrow-\infty}(1)+\lim _{t \rightarrow-\infty}\frac{\cos t}{t}}\\ &=\frac{0-1+0}{1+0}\\ &=-1 \end{align*}