To evaluate the limit

$\lim_{x \to \frac{\pi}{2}^-} \frac{\tan x}{\frac{9}{2}x - \pi},$

we need to determine the behavior of the numerator and denominator as $x$ approaches $\frac{\pi}{2}$ from the left.

1. Numerator: $\tan x$ approaches $\infty$ as $x$ approaches $\frac{\pi}{2}$.
2. Denominator: $\frac{9}{2}x - \pi$ approaches $\frac{9}{2} \cdot \frac{\pi}{2} - \pi = \frac{9\pi}{4} - \pi = \frac{5\pi}{4}$, which is a finite non-zero value.

Since the limit evaluates to the form $\frac{\infty}{\text{finite}}$, we can conclude that the limit is $\infty$.

Given this, the best approach for evaluating this limit is:

Choice A: Use l'Hôpital's Rule to rewrite the limit.

Conclusion

Thus, the answer is:

A. Use l'Hôpital's Rule to rewrite the limit as

$\lim_{x \to \frac{\pi}{2}^-} \left(\text{something}\right).$

Would you like more details on using l'Hôpital's Rule or any other questions? Here are some related questions you might consider:

1. What are the conditions under which l'Hôpital's Rule can be applied?
2. How do you differentiate $\tan x$ and $\frac{9}{2}x - \pi$?
3. What other methods could be used to evaluate limits involving trigonometric functions?
4. How can we verify the limit using a graph?
5. What is the significance of approaching a limit from the left versus the right?

Tip: When faced with indeterminate forms, always check if l'Hôpital's Rule is applicable or if simplification can be done beforehand.