$ \lim_{\theta \rightarrow 0^+} \cos \left ( 1 \over \theta \right ) \left ( \sin \theta - \theta \over \theta \right) $ Let, $ \frac{1}{\theta} = \phi $, then we have for all real values of $ \phi $ $ -1 \leq \left ( \cos \phi \right ) \leq 1 $
EDIT:: taking comments into consideration $ - \left| \dfrac{\sin \theta}{\theta} - 1 \right| \le \cos(1/\theta) \left(\dfrac{\sin \theta}{\theta} - 1 \right) \le \left|\dfrac{\sin \theta}{\theta} - 1 \right| $ $ -1 \times 0 \leq \lim_{\phi \rightarrow \infty } \cos \phi \times \lim_{\theta \rightarrow 0 } \left ( \frac{\sin \theta} \theta - 1 \right ) \leq 1 \times 0 $ $ \text{Or, using squeeze theorem, we have, } \lim_{\theta \rightarrow 0^+} \cos \left ( 1 \over \theta \right ) \left ( \sin \theta - \theta \over \theta \right) = 0 $