Answer
$\lim\limits_{x\to\frac{1}{2}^-}(x\sec{(\pi x)})=+\infty.$
Work Step by Step
$\lim\limits_{x\to\frac{1}{2}^-}(x\sec{(\pi x)})\to$
$\lim\limits_{x\to\frac{1}{2}^-}(x)=\dfrac{1}{2}^-=\dfrac{1}{2}.$
$\lim\limits_{x\to\frac{1}{2}^-}\sec{(\pi x)}=\sec{\frac{1}{2}^-(\pi)}=+\infty.$
By Theorem $1.15:$
$\lim\limits_{x\to\frac{1}{2}^-}(x\sec{(\pi x)})=+\infty.$
