Answer
$\frac{1}{7}$
Work Step by Step
$\lim\limits_{\theta \to 0}\frac{\sin \theta}{\tan 7\theta}=\lim\limits_{\theta \to 0}\frac{1}{7}\cdot \frac{\sin \theta}{\theta}\cdot \frac{7\theta}{\tan 7\theta}$ (Use the properties of limits)
$=\frac{1}{7}\lim\limits_{\theta \to 0}\frac{\sin \theta}{\theta}\cdot \lim\limits_{\theta \to 0}\frac{7\theta}{\tan 7\theta}$ (As $\theta\rightarrow 0$, $7\theta\rightarrow 0$)
$=\frac{1}{7}\lim\limits_{\theta \to 0}\frac{\sin \theta}{\theta}\cdot \lim\limits_{7\theta \to 0}\frac{7\theta}{\tan 7\theta}$
$=\frac{1}{7}\cdot 1\cdot 1$
$=\frac{1}{7}$
