Find the one-sided limit (if it exists). (If the limit is infinite, enter 'oo or \( { }^{\prime}-\infty \), as appropriate. If the limit does not otherwise exist, enter D \[ \lim _{x \rightarrow(1 / 2)^{-}} 4 x \sec (\pi x) \]

To find the one-sided limit \[ \lim_{x \rightarrow (1/2)^{-}} 4x \sec(\pi x), \] we start by substituting \( x \) approaching \( \frac{1}{2} \) from the left. 1. **Evaluate \( \sec(\pi x) \)**: As \( x \rightarrow \frac{1}{2} \), we have \( \pi x \rightarrow \frac{\pi}{2} \). The secant function is defined as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). At \( \theta = \frac{\pi}{2} \), \( \cos\left(\frac{\pi}{2}\right) = 0 \). Therefore, \( \sec(\frac{\pi}{2}) \) approaches infinity (\( \infty \)). 2. **Consider the multiplication by \( 4x \)**: As \( x \) approaches \( \frac{1}{2} \) from the left, \( 4x \) approaches \( 4 \cdot \frac{1}{2} = 2 \). Now, combining these results: \[ 4x \sec(\pi x) \rightarrow 2 \cdot \infty = \infty. \] Thus, the limit is: \[ \lim_{x \rightarrow (1/2)^{-}} 4x \sec(\pi x) = \infty. \] In conclusion, the one-sided limit exists and is \(\oo\).