Answer
The limit does not exist.
Work Step by Step
We have
$$
\lim _{\theta \rightarrow 0} \frac{\cos \theta -2}{\theta}=\frac{1-2}{0}=\frac{-1}{0}.
$$
The one-sided limits are
$$
\lim _{\theta \rightarrow 0^-} \frac{\cos \theta -2}{\theta}=\frac{1-2}{0^-}=\frac{-1}{0^-}=\infty.
$$
$$
\lim _{\theta \rightarrow 0^+} \frac{\cos \theta -2}{\theta}=\frac{1-2}{0^+}=\frac{-1}{0^+}=-\infty.
$$
So the limit does not exist.
