Why is $\lim_{x\to\frac\pi2^+}\frac7x\sec x=-\infty$? Why is the below true?
$$\lim_{x\to\frac\pi2^+}\frac7x\sec x=-\infty$$
Do I need to know the graph of the secant function?

You don't necessarily need to know the graph of secant.
Recall that $\sec(x)=\frac{1}{\cos(x)}$.
Thus:
$$\frac{7}{x}\sec(x)=\frac{7}{x\cos(x)}$$
Using direct substitution:
$$\lim_{x\to\frac{\pi}{2}^+}\frac{7}{x\cos(x)}=\frac{14}{\pi}\cdot\frac{1}{\cos(\pi/2)}=\frac{14}{\pi}\cdot\frac{-1}{0}=-\infty$$
Since we are approaching $\pi/2$ from the right, we are approaching $\pi/2$ from the second quadrant, ergo the second expression becomes negative.
