Answer
$$\frac{2}{3}$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\theta \to 0} 2\theta \cot 3\theta \cr
& {\text{trigonometric identity cot}}\alpha = \frac{{\cos \alpha }}{{\sin \alpha }} \cr
& = \mathop {\lim }\limits_{\theta \to 0} 2\theta \left( {\frac{{cos3\theta }}{{\sin 3\theta }}} \right) \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{2\theta cos3\theta }}{{\sin 3\theta }} \cr
& {\text{evaluating the limit}} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{2\theta cos3\theta }}{{\sin 3\theta }} = \frac{0}{0} \cr
& {\text{applying l'Hopital's rule}} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{\frac{d}{{d\theta }}\left[ {2\theta cos3\theta } \right]}}{{\frac{d}{{d\theta }}\left[ {\sin 3\theta } \right]}} \cr
& {\text{use product rule in the numerator}} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{2\theta \frac{d}{{d\theta }}\left[ {cos3\theta } \right] + \cos 3\theta \frac{d}{{d\theta }}\left[ {2\theta } \right]}}{{\frac{d}{{d\theta }}\left[ {\sin 3\theta } \right]}} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{2\theta \left( { - 3\sin 3\theta } \right) + 2\cos 3\theta }}{{3\cos 3\theta }} \cr
& = \mathop {\lim }\limits_{\theta \to 0} \frac{{ - 6\theta \sin 3\theta + 2\cos 3\theta }}{{3\cos 3\theta }} \cr
& {\text{evaluating the limit}} \cr
& = \frac{{ - 6\left( 0 \right)\sin 3\left( 0 \right) + 2\cos 3\left( 0 \right)}}{{3\cos 3\left( 0 \right)}} \cr
& = \frac{2}{3} \cr} $$
