[tex]\displaystyle\lim_{\theta\to0}\frac{\cos4\theta-1}{\sin7\theta}[/tex]
[tex]\displaystyle\lim_{\theta\to0}\frac{(4\theta)(7\theta)(\cos4\theta-1)}{(4\theta)(7\theta)\sin7\theta}[/tex]
[tex]\displaystyle\lim_{\theta\to0}\frac{4\theta}{7\theta}\cdot\lim_{\theta\to0}\frac{\cos4\theta-1}{4\theta}\cdot\lim_{\theta\to0}\frac{7\theta}{\sin7\theta}[/tex]
The second limit is 0, so the original limit is also 0.
