How do I go about proving that $\lim_{x\rightarrow1/2}{2x\sec(πx)}=\infty$ with some type of rigor? I'm rusty.
You are talking about the left handed limit $\displaystyle \lim_{x \to \frac12^-} \frac{2x}{\cos \pi x}=+\infty$
Because $\displaystyle \cos \tfrac\pi2 =0, \, 1 \ne 0$ and $\displaystyle \cos \pi x > 0, \, 2x > 0$ for $\displaystyle 0 < x < \tfrac12$.
Prove that for every $\varepsilon>0$ there exists $\delta>0$ such that $x\in \left(\dfrac{1}{2}-\delta,\dfrac{1}{2}\right)$, you have $2x\sec(\pi x)>\varepsilon$