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The problem asks to determine the infinite limit:

\[
\lim_{x \to -\frac{3}{2}^{-}} x \sec(\pi x)
\]

### Step-by-step solution:

1. **Understanding the behavior of \(\sec(\pi x)\)**:
   - Recall that \(\sec(\theta) = \frac{1}{\cos(\theta)}\), so \(\sec(\pi x) = \frac{1}{\cos(\pi x)}\).
   - The cosine function, \(\cos(\pi x)\), is periodic with a period of \(2\). Let's check what happens when \(x = -\frac{3}{2}\):
     \[
     \cos\left(\pi \times \left(-\frac{3}{2}\right)\right) = \cos\left(-\frac{3\pi}{2}\right) = 0.
     \]
     Since \(\cos\left(-\frac{3\pi}{2}\right) = 0\), it implies that \(\sec\left(\pi x\right)\) approaches infinity as \(x \to -\frac{3}{2}^{-}\), because \(\sec(\pi x) = \frac{1}{0}\) leads to vertical asymptotes.

2. **Limit behavior near \(x = -\frac{3}{2}\)**:
   - As \(x\) approaches \(-\frac{3}{2}\) from the left (\(x \to -\frac{3}{2}^{-}\)), \(\sec(\pi x)\) approaches \(-\infty\) because the cosine value goes to \(0\) from the negative side.
   - Therefore, we have:
     \[
     \lim_{x \to -\frac{3}{2}^{-}} \sec(\pi x) = -\infty.
     \]
   - Also, \(x\) approaches \(-\frac{3}{2}\) from the left, meaning \(x < -\frac{3}{2}\), so \(x\) is a negative number near this value.

3. **Multiplying by \(x\)**:
   - The term \(x \sec(\pi x)\) now behaves as:
     \[
     x \times (-\infty) \quad \text{as} \quad x \to -\frac{3}{2}^{-}.
     \]
   - Since \(x\) is negative and \(\sec(\pi x)\) tends to \(-\infty\), multiplying a negative number by \(-\infty\) gives \(+\infty\).

4. **Conclusion**:
   \[
   \lim_{x \to -\frac{3}{2}^{-}} x \sec(\pi x) = +\infty.
   \]

Let me know if you'd like further details or explanations on this.

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Here are five related questions to extend your understanding:
1. What is the general behavior of \(\sec(\theta)\) as \(\theta\) approaches values where \(\cos(\theta) = 0\)?
2. How does the sign of \(x\) affect the result in infinite limits when multiplying by a divergent function?
3. What happens to \(\sec(\pi x)\) when \(x\) approaches other values like \(0\) or \(1\)?
4. Can you determine the limit of \(x \tan(\pi x)\) as \(x \to -\frac{3}{2}^{-}\)?
5. How would the limit change if the cosine term was replaced by a sine or tangent function?

**Tip:** When evaluating limits involving trigonometric functions, always check for points where the denominator might become zero, leading to asymptotes or undefined behavior.

Determine the infinite limit: lim(x→-3/2^-) x sec(πx)

Learn how to determine the infinite limit lim(x→-3/2^-) x sec(πx) with a detailed step-by-step solution. This problem covers key calculus concepts, including the behavior of trigonometric functions and their limits. Suitable for high school or early college students studying calculus.

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