Answer
$$\frac{3}{7}$$
Work Step by Step
Given $$
\lim _{x \rightarrow \pi / 2^{-}} \sec 7 x \cos 3 x
$$
Then
\begin{align*}
\lim _{x \rightarrow \pi / 2^{-}} \sec 7 x \cos 3 x&=\lim _{x \rightarrow \pi / 2^{-}} \frac{\cos 3 x}{\cos 7 x}, \ \ \ \text{Using L'Hopital's rule } \\
&=\lim _{x \rightarrow \pi / 2^{-}} \frac{-3\sin 3 x}{-7\sin 7 x}\\
&=\frac{3}{7}
\end{align*}
