Answer
$$0$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{t \to 0} \frac{{1 - \cos 6t}}{{2t}} \cr
& {\text{evaluating the limit}} \cr
& = \frac{{1 - \cos 6\left( 0 \right)}}{{2\left( 0 \right)}} \cr
& = \frac{{1 - 1}}{{2\left( 0 \right)}} \cr
& = \frac{0}{0} \cr
& \cr
& {\text{applying l'Hopital's rule}} \cr
& = \mathop {\lim }\limits_{t \to 0} \frac{{\frac{d}{{dt}}\left[ {1 - \cos 6t} \right]}}{{\frac{d}{{dt}}\left[ {2t} \right]}} \cr
& = \mathop {\lim }\limits_{t \to 0} \frac{{6\sin 6t}}{2} \cr
& = \mathop {\lim }\limits_{t \to 0} 3\sin 6t \cr
& {\text{evaluating the limit}} \cr
& = \mathop {\lim }\limits_{t \to 0} 3\sin 6t = 3\sin 6\left( 0 \right) \cr
& = 0 \cr} $$
