Answer
$$1$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\theta \to 0} \frac{{3\sin 8\theta }}{{8\sin 3\theta }} \cr
& {\text{evaluating the limit}} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{3\sin 8\left( 0 \right)}}{{8\sin 3\left( 0 \right)}} = \frac{0}{0} \cr
& {\text{applying l'Hopital's rule}} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{\frac{d}{{d\theta }}\left[ {3\sin 8\theta } \right]}}{{\frac{d}{{d\theta }}\left[ {8\sin 3\theta } \right]}} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{24\cos 8\theta }}{{24\cos 3\theta }} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{\cos 8\theta }}{{\cos 3\theta }} \cr
& {\text{evaluating the limit}} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{\cos 8\theta }}{{\cos 3\theta }} = \frac{{\cos 8\left( 0 \right)}}{{\cos 3\left( 0 \right)}} \cr
& = 1 \cr} $$