Answer
$$\lim _{\theta \rightarrow -\infty} \frac{\cos \theta }{3\theta } =0$$
Work Step by Step
Since $-1 \leq \cos \theta \leq 1$ for all $\theta \in \mathbb{R}$ then
$$\frac{-1}{3\theta} \leq \frac{\cos \theta }{3\theta } \leq \frac{1}{3\theta }$$
for all $\theta \in \mathbb{R} \backslash\{0\}$ . Hence since $$\lim _{\theta \rightarrow- \infty}\frac{-1}{3\theta}=0=\lim _{\theta \rightarrow- \infty}\frac{ 1}{3\theta}$$ we have by the sandwich theorem that
$$\lim _{\theta \rightarrow -\infty} \frac{\cos \theta }{3\theta } =0$$
