https://r2.erweima.ai/mathbot/image/1726432515701m76erwof.jpg

The problem involves analyzing the limits of the functions \( f(x) \), \( g(x) \), and \( h(x) \), where \( h(x) \) satisfies the condition \( f(x) < h(x) < g(x) \) for all \( x \neq 1 \).

### Step-by-step Solution:

1. **Functions \( f(x) \) and \( g(x) \):**
   \[
   f(x) = 
   \begin{cases} 
   (-(x-1))^2 + 3 = (x-1)^2 + 3, & \text{if } x \neq 1 \\
   -2, & \text{if } x = 1 
   \end{cases}
   \]
   \[
   g(x) = 
   \begin{cases} 
   ((x-1))^2 + 3, & \text{if } x \neq 1 \\
   -2, & \text{if } x = 1 
   \end{cases}
   \]
   
   For \( x \neq 1 \), both functions \( f(x) \) and \( g(x) \) are the same: \( (x-1)^2 + 3 \). So, when taking the limit as \( x \to 1 \), we evaluate the behavior of \( (x-1)^2 + 3 \) at \( x = 1 \).

2. **Limit of \( f(x) \) and \( g(x) \) as \( x \to 1 \):**
   \[
   \lim_{x \to 1} f(x) = \lim_{x \to 1} g(x) = \lim_{x \to 1} ((x-1)^2 + 3) = 3.
   \]

   Therefore:
   \[
   \lim_{x \to 1} f(x) = 3, \quad \lim_{x \to 1} g(x) = 3.
   \]

3. **Behavior of \( h(x) \):**
   Since \( f(x) < h(x) < g(x) \) for \( x \neq 1 \), the limit of \( h(x) \) as \( x \to 1 \) must also be squeezed between \( f(x) \) and \( g(x) \). Hence, by the Squeeze Theorem:
   \[
   \lim_{x \to 1} h(x) = 3.
   \]

4. **Given limit expressions:**

   - \(\lim_{x \to 1} (4h(x) - 1) = \lim_{x \to 1} (4 \cdot 3 - 1) = 12 - 1 = 11\).
   
   The correct limit is \( \lim_{x \to 1} (4h(x) - 1) = 11 \), not -9.

### Conclusion:
- The correct option is: **\( \lim_{x \to 1} (4h(x) - 1) = 11 \)**.

Would you like a further explanation on any of these steps or the Squeeze Theorem?

---

Here are 5 related questions to expand your understanding:

1. What is the Squeeze Theorem, and how is it applied in this context?
2. How do the functions \( f(x) \) and \( g(x) \) behave for values of \( x \neq 1 \)?
3. Why is the limit at \( x = 1 \) different for \( f(x) \) and \( g(x) \) when \( x = 1 \)?
4. How does the continuity of \( f(x) \) and \( g(x) \) influence the limit of \( h(x) \)?
5. What would happen if \( h(x) \) did not satisfy the inequality \( f(x) < h(x) < g(x) \)?

**Tip:** Always check for conditions like \( x \neq 1 \) or continuity before calculating limits at specific points.

Analyze the limits of the functions f(x), g(x), and h(x), where h(x) satisfies f(x) < h(x) < g(x) for all x ≠ 1. Evaluate the limits, including lim (4h(x) - 1) as x approaches 1.

Explore the limit analysis of functions f(x), g(x), and h(x) as x approaches 1, using the Squeeze Theorem. This problem involves limits of piecewise functions and provides a detailed solution on evaluating expressions like lim (4h(x) - 1) and understanding the behavior of functions near discontinuities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation