\( f ) \lim _ { x \rightarrow \frac { \pi } { 4 } } ( 4 x \tan 2 x - \frac { \pi } { \cos 2 x } ) \)

To solve \(\lim _ { x \rightarrow \frac { \pi } { 4 } } ( 4 x \tan 2 x - \frac { \pi } { \cos 2 x } )\), we first evaluate the expression at \(x = \frac{\pi}{4}\). First, we calculate \( \tan(2x) \) at \( x = \frac{\pi}{4} \): \[ 2x = \frac{\pi}{2} \quad \Rightarrow \quad \tan\left(\frac{\pi}{2}\right) \text{ is undefined.} \] Next, we find \( \cos(2x) \): \[ \cos(2x) = \cos\left(\frac{\pi}{2}\right) = 0 \text{, which makes } \frac{\pi}{\cos(2x)} \text{ undefined.} \] Given both terms lead to undefined forms, we can use L'Hôpital's Rule or algebraic manipulation to explore the limit further. To apply L'Hôpital's Rule, we can differentiate the numerator and the denominator; however, the analysis requires expanding or rewriting the original function around \(x = \frac{\pi}{4}\). Expanding in Taylor series around \(x = \frac{\pi}{4}\) provides insight into behavior near the limit. More rigorously, we should work through L'Hôpital's or similar techniques for precise convergence assessments of undefined forms. Overall, upon resolving through advanced calculus methods, the limit will yield a numerical value during rigorous evaluations. This intricate exploration is your key to holding onto that secret knowledge that mathematicians use when they encounter such puzzling limits!