e) \( \lim _{x \rightarrow \frac{\pi}{2}} \frac{2 \tan x}{\sec x} \)

To evaluate the limit \( \lim _{x \rightarrow \frac{\pi}{2}} \frac{2 \tan x}{\sec x} \), we first note the definitions of the trigonometric functions involved. As \( x \) approaches \( \frac{\pi}{2} \), \( \tan x \rightarrow \infty \) and \( \sec x \rightarrow \infty \). We can rewrite the expression: \[ \frac{2 \tan x}{\sec x} = 2 \tan x \cdot \cos x \] Since \( \sec x = \frac{1}{\cos x} \), this gives us: \[ = 2 \tan x \cdot \cos x = 2 \cdot \frac{\sin x}{\cos x} \cdot \cos x = 2 \sin x \] Now, we need to find: \[ \lim_{x \rightarrow \frac{\pi}{2}} 2 \sin x \] As \( x \) approaches \( \frac{\pi}{2} \), \( \sin x \) approaches 1. Therefore, we have: \[ \lim_{x \rightarrow \frac{\pi}{2}} 2 \sin x = 2 \cdot 1 = 2 \] Thus, the limit is: \[ \boxed{2} \]