1. **Stating the problem:** We want to evaluate the limit $$\lim_{x\to \frac{\pi}{2}} x \sec^2 x$$. 2. **Recall the definition:** $$\sec x = \frac{1}{\cos x}$$, so $$\sec^2 x = \frac{1}{\cos^2 x}$$. 3. **Examine behavior near $$x = \frac{\pi}{2}$$:** - As $$x \to \frac{\pi}{2}$$, $$\cos x \to 0$$. - Thus, $$\sec^2 x = \frac{1}{\cos^2 x} \to \infty$$ or $$-\infty$$ depending on direction. 4. **Determine the sign of $$\sec^2 x$$ near $$\frac{\pi}{2}$$:** - Since $$\cos^2 x \geq 0$$ for all $$x$$ and is squared, $$\sec^2 x = \frac{1}{\cos^2 x}$$ is always positive (or tends to positive infinity). 5. **Evaluate the factor $$x$$ at the limit point:** - As $$x \to \frac{\pi}{2} \approx 1.5708$$, $$x$$ approaches a finite positive number. 6. **Combine:** - The expression $$x \sec^2 x = x \cdot \frac{1}{\cos^2 x}$$ - Since $$x$$ is finite positive near $$\frac{\pi}{2}$$ and $$\sec^2 x \to +\infty$$, their product tends to $$+\infty$$. 7. **Conclusion:** $$\boxed{\lim_{x\to \frac{\pi}{2}} x \sec^2 x = +\infty}$$