Answer
The limit is 0.
Work Step by Step
Limit (L)
$= \lim\limits_{\theta \to 0} \frac{\cos{\theta}-1}{\sin{\theta}}$
$= \lim\limits_{\theta \to 0} \frac{\cos{\theta}-1}{\theta}
\times \frac{\theta}{\sin{\theta}}$
$= \lim\limits_{\theta \to 0} \frac{\cos{\theta}-1}{\theta}
\times \lim\limits_{\theta \to 0} \frac{\theta}{\sin{\theta}}$
By equation 3:
$$\lim\limits_{\theta \to 0} \frac{\cos{\theta}-1}{\theta}=0$$
And by equation 2:
$$\lim\limits_{\theta \to 0} \frac{\sin{\theta}}{\theta}=1$$
Thus,
L $=0\times1^{-1}=0\times1=0$
Therefore, the limit is 0.