\[\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \left( {\operatorname{tg} 3x\operatorname{tg} 8x} \right)\]
\[L = \mathop {\lim }\limits_{t \to 0} \left( { - \operatorname{ctg} 3t\operatorname{tg} 8t} \right) = - \mathop {\lim }\limits_{t \to 0} \left( {\frac{{\operatorname{tg} 8t}}{{\operatorname{tg} 3t}}} \right) = - \frac{8}{3}.\]
\[ - \frac{8}{3}\]
