In the graph of $y = \tan(x)$, the behavior of the function as $x \to \frac{\pi}{2}^+$ can be analyzed using the properties of the tangent function.

Key Insight:

• The tangent function, $\tan(x) = \frac{\sin(x)}{\cos(x)}$, has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$ (where $n \in \mathbb{Z}$).
• Near $x = \frac{\pi}{2}^+$, the denominator $\cos(x)$ approaches $0$ from the negative side (since $\cos(x)$ transitions from positive to negative as $x$ crosses $\frac{\pi}{2}$).
• As a result, $\tan(x) \to -\infty$.

Conclusion:

As $x \to \frac{\pi}{2}^+$, $\tan(x) \to -\infty$.

Would you like more details on the tangent function's asymptotic behavior or related concepts?

Here are some related questions to explore:

1. What happens to $\tan(x)$ as $x \to \frac{\pi}{2}^-$?
2. What is the period of the tangent function?
3. Why does $\tan(x)$ have vertical asymptotes at $x = \frac{\pi}{2} + n\pi$?
4. How is the tangent function related to the unit circle?
5. What are the key differences between $\tan(x)$ and $\sin(x)$ or $\cos(x)$?

Tip: Remember that the tangent function is undefined at points where $\cos(x) = 0$, leading