For [tex]x=\dfrac{1}{2}[/tex] the function is undefined.
[tex]\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)
[/tex]
So, the limit doesn't exist.
