Answer
$3$
Work Step by Step
Consider: $\lim\limits_{\theta \to \dfrac{-\pi}{3}}f(\theta)=\lim\limits_{ \theta \to \dfrac{-\pi}{3}}\dfrac{3 \theta +\pi}{\sin (\theta+\dfrac{\pi}{3})}$
Now, $f(\dfrac{-\pi}{3})=\dfrac{3 (\dfrac{-\pi}{3}) +\pi}{\sin ((\dfrac{-\pi}{3})+\dfrac{\pi}{3})}=\dfrac{0}{0}$
The limit shows an indeterminate form. Thus, apply L-Hospital's rule: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$
Then:
$\lim\limits_{\theta \to \dfrac{-\pi}{3}}\dfrac{3}{\cos (\theta+\dfrac{\pi}{3})}=\dfrac{3}{\cos (\dfrac{-\pi}{3}+\dfrac{\pi}{3})}=\dfrac{3}{\cos 0}=\dfrac{3}{1}=3 $
