Answer
$$\lim_{\theta\to-\infty}\frac{\cos\theta}{3\theta}=0$$
Work Step by Step
We know already that $$-1\le \cos\theta\le1$$
So, $$-\frac{1}{3\theta}\le\frac{\cos\theta}{3\theta}\le\frac{1}{3\theta}$$
As $\theta\to-\infty$, $3\theta$ also approaches $-\infty$, and both $1/3\theta$ and $-1/3\theta$ will approach $0$. So $\lim_{\theta\to-\infty}(-1/3\theta)=\lim_{\theta\to\infty}(1/3\theta)=0$
Therefore, according to Squeeze Theorem, we conclude that $$\lim_{\theta\to-\infty}\frac{\cos\theta}{3\theta}=0$$
