ModifyingBelow lim With x right arrow StartFraction pi Over 2 EndFraction Superscript minus
StartStartFraction tangent x OverOver left parenthesis StartFraction 9 Over 2 x minus pi EndFraction right parenthesis EndEndFraction
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Part 1
How should the given limit be​ evaluated? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
Use l'Hôpital's Rule to rewrite the limit as ModifyingBelow lim With x right arrow StartFraction pi Over 2 EndFraction Superscript minus
left parenthesis nothing right parenthesis

.
B.
Use a trigonometric identity to rewrite the limit as ModifyingBelow lim With x right arrow StartFraction pi Over 2 EndFraction Superscript minus
left parenthesis nothing right parenthesis

.
C.
Multiply the expression by a unit fraction to obtain ModifyingBelow lim With x right arrow StartFraction pi Over 2 EndFraction Superscript minus
left parenthesis nothing right parenthesis

.
D.
Use direct substitution.

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.

ModifyingBelow lim With x right arrow StartFraction pi Over 2 EndFraction Superscript minus StartStartFraction tangent x OverOver left parenthesis StartFraction 9 Over 2 x minus pi EndFraction right parenthesis EndEndFraction

Discover how to evaluate the limit of the function (tan x) / (9/2 x - pi) as x approaches π/2 from the left. This solution highlights the application of L'Hôpital's Rule and addresses the behavior of trigonometric functions at critical points. Ideal for students in Grades 11-12.

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