Answer
$0$
Work Step by Step
\begin{align*}
\lim _{\theta\rightarrow 0} \frac{ \sec \theta-1 }{\theta }&= \lim _{\theta\rightarrow 0} \frac{ 1- \cos \theta }{\theta } \frac{1}{\cos \theta}\\
&= \lim _{\theta\rightarrow 0} \frac{ 1- \cos \theta }{\theta } \lim _{\theta\rightarrow 0}
\frac{1}{\cos \theta}\\
&=0.
\end{align*}
Where we used the fact that $\lim _{\theta\rightarrow 0} \frac{ 1- \cos \theta }{\theta } =0.$
