https://file.aiquickdraw.com/mathbot/file/1725697004602jsvqy0ft.jpg

Let's analyze the function and the corresponding questions step by step based on the image.

The given function \( f(x) \) is a piecewise function:

\[
f(x) = \begin{cases}
8x^2 & \text{if } x < -1 \\
8x & \text{if } -1 \leq x < 1 \\
\frac{8}{x} & \text{if } x \geq 1
\end{cases}
\]

You are asked to find the following values:

1. **\( f(-1) = -8 \)**:  
   This is correct. For \( x = -1 \), we use the second part of the function \( f(x) = 8x \), so:
   \[
   f(-1) = 8(-1) = -8
   \]
   
2. **\( f(1) = 8 \)**:  
   This is correct. For \( x = 1 \), we use the third part of the function \( f(x) = \frac{8}{x} \), so:
   \[
   f(1) = \frac{8}{1} = 8
   \]

3. **\( \lim_{x \to -1^-} f(x) = -\infty \)**:  
   This value is marked incorrect. Let's evaluate:
   - As \( x \to -1 \) from the left, we use the first part of the function \( f(x) = 8x^2 \).
   - As \( x \to -1 \), we get:
     \[
     \lim_{x \to -1^-} 8x^2 = 8(-1)^2 = 8
     \]
   So, the correct answer should be 8, not \( -\infty \). The marked answer of \( -\infty \) is indeed incorrect.

4. **\( \lim_{x \to -1^+} f(x) = -8 \)**:  
   This is correct. As \( x \to -1^+ \), we use the second part of the function \( f(x) = 8x \), and:
   \[
   \lim_{x \to -1^+} 8x = 8(-1) = -8
   \]

5. **\( \lim_{x \to 1^-} f(x) = 8 \)**:  
   This is correct. As \( x \to 1^- \), we use the second part of the function \( f(x) = 8x \), so:
   \[
   \lim_{x \to 1^-} 8x = 8(1) = 8
   \]

6. **\( \lim_{x \to 1^+} f(x) = 8 \)**:  
   This is correct. As \( x \to 1^+ \), we use the third part of the function \( f(x) = \frac{8}{x} \), and:
   \[
   \lim_{x \to 1^+} \frac{8}{x} = \frac{8}{1} = 8
   \]

---

Would you like further details on any of these solutions?

Here are five related questions for further exploration:

1. What happens to the limit \( \lim_{x \to -1^-} f(x) \) for functions with non-quadratic terms?
2. How can piecewise continuity at points like \( x = -1 \) and \( x = 1 \) be verified mathematically?
3. What would the graph of this piecewise function look like?
4. How does the behavior of the function change for large values of \( x \)?
5. How would you approach finding the derivative of such a piecewise function?

**Tip**: When evaluating limits in piecewise functions, always check the behavior from both sides of the point in question to verify if the limit exists and is finite.

Find the values for f(-1), f(1), and limits for a piecewise function with three parts: 8x^2 if x < -1, 8x if -1 ≤ x < 1, and 8/x if x ≥ 1.

Explore how to solve a piecewise function with three distinct parts and compute limits at the boundary points. This problem involves concepts such as limits, continuity, and piecewise functions, suitable for students in Grades 11-12.

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