Question

I need to find the limit

Answer 1

For $x=\dfrac{1}{2}$ the function is undefined.

$\displaystyle \lim_{x\to\tfrac{1}{2}^-} x\sec (\pi x)=\infty\\ \lim_{x\to\tfrac{1}{2}^+} x\sec (\pi x)=-\infty\\\\ \lim_{x\to\tfrac{1}{2}^-}x\sec (\pi x)\not = \lim_{x\to\tfrac{1}{2}^+}x\sec (\pi x)$
So, the limit doesn't exist.