Complicated trig limits to infinity problem!?

Question

$$\lim_{t\to-\infty}\frac{2-t+\sin t}{t+\cos t}$$

I don't know how to do this problem! I don't think you can use the squeeze theorem on it? Any help would be appreciated.

Answer 1

HINT: $|2+\sin t|\le 3$ and $|\cos t|\le 1$ for all $t$, so for $t$ of large absolute value the fraction is approximately $\dfrac{-t}t$.

Answer 2

$$\lim_{t\to-\infty}\frac{2-t+\sin t}{t+\cos t}=\lim_{t\to-\infty}\frac{\frac2t-1+\frac{\sin t}{t}}{1+\frac{\cos t}{t}}=-1.$$

Why $\frac{\sin t}{t}\to 0$ and $\frac{\cos t}{t} \to 0$ as $t\to -\infty $?

You can use the squeeze theorem to get it.

Hints: As $t\to -\infty$, $\frac1t\le \frac{\sin t}{t}\le -\frac1t$.