Question:

Step 1: Substitute $x = \frac{\pi}{2} + h$ where $h \to 0$ to simplify the limit. This substitution transforms the limit as $x \to \frac{\pi}{2}$ into a limit as $h \to 0$, making it easier to work with small quantities. With this substitution: $$x - \pi = \frac{\pi}{2} + h - \pi = h - \frac{\pi}{2}$$ $$\pi - 2x = \pi - 2\left(\frac{\pi}{2} + h\right) = \pi - \pi - 2h = -2h$$ $$x - \frac{\pi}{2} = h$$ Step 2: Simplify $\cos^2 x$ using the substitution. We have: $$\cos x = \cos\left(\frac{\pi}{2} + h\right) = -\sin h$$ For small $h$, using the approximation $\sin h \approx h$: $$\cos x \approx -h$$ $$\cos^2 x \approx h^2$$ Step 3: Simplify $\tan\left(x - \frac{\pi}{2}\right)$ using the substitution. $$\tan\left(x - \frac{\pi}{2}\right) = \tan(h)$$ For small $h$, using the approximation $\tan h \approx h$: $$\tan(h) \approx h$$ Step 4: Substitute the approximations into the original expression. The numerator becomes: $$4(x - \pi)\cos^2 x \approx 4\left(h - \frac{\pi}{2}\right) \cdot h^2$$ As $h \to 0$, we have $h - \frac{\pi}{2} \to -\frac{\pi}{2}$, so: $$\text{Numerator} \approx 4 \cdot \left(-\frac{\pi}{2}\right) \cdot h^2 = -2\pi h^2$$ The denominator becomes: $$\pi(\pi - 2x)\tan\left(x - \frac{\pi}{2}\right) \approx \pi \cdot (-2h) \cdot h = -2\pi h^2$$ Step 5: Evaluate the limit by dividing numerator by denominator. $$\lim_{h \to 0} \frac{-2\pi h^2}{-2\pi h^2} = \frac{-2\pi h^2}{-2\pi h^2} = 1$$ The $h^2$ terms cancel, and the constant coefficients simplify to give us 1. **Final Answer:** The value of the limit is $\boxed{1}$, which corresponds to **Option 1**.