https://file.aiquickdraw.com/mathbot/file/1725982124723juv20px0.jpg

Let's analyze the problem based on the image you have provided.

We are asked to determine the limits of the function \( f(x) \) for different values of \( x \), using the graph.

### Here's a breakdown of the graph:

- The graph of the function is given with specific behavior as \( x \) approaches certain values. Based on the instructions, the limits are:

#### (a) \( \lim_{x \to -8} f(x) \)
- Looking at the graph near \( x = -8 \), there is a hole in the graph, meaning that the function approaches a specific value from both sides but does not actually reach it. From the image, it appears that as \( x \) approaches \(-8\), the \( y \)-value approaches \(-2\). Therefore, \( \lim_{x \to -8} f(x) = -2 \).

#### (b) \( \lim_{x \to 2^-} f(x) \)
- For the left-hand limit as \( x \) approaches 2, look at the graph as \( x \) approaches 2 from values less than 2. The graph seems to approach \( y = 0 \) from the left. So, \( \lim_{x \to 2^-} f(x) = 0 \).

#### (c) \( \lim_{x \to 2^+} f(x) \)
- For the right-hand limit as \( x \) approaches 2, observe how the function behaves from the right of \( x = 2 \). The function appears to approach 2, but there's no continuity here. Hence, this limit might not exist. The correct answer is likely "none" for \( \lim_{x \to 2^+} f(x) \).

#### (d) \( \lim_{x \to \infty} f(x) \)
- As \( x \to \infty \), the graph shows that the function increases and heads towards \( y = 18 \), so the correct value for this limit is 18.

### Final answers:
- (a) \( \lim_{x \to -8} f(x) = -2 \)
- (b) \( \lim_{x \to 2^-} f(x) = 0 \)
- (c) \( \lim_{x \to 2^+} f(x) = \text{none} \)
- (d) \( \lim_{x \to \infty} f(x) = 18 \) (correct)

Let me know if you'd like more details or have questions about this problem.

#### Related Questions:
1. What are left-hand and right-hand limits, and why are they important?
2. How do you interpret a limit when there's a hole in the graph at a specific point?
3. What does it mean when a limit "does not exist"?
4. How does behavior at infinity affect the interpretation of limits?
5. What are the differences between continuous and discontinuous functions in limit calculations?

#### Tip:
Always check for continuity and jumps in the graph at the point you're evaluating for the limit. These give clues about whether the limit exists or not!

Use the figure below, which gives a graph of the function f(x), to give values for the indicated limits. If a limit does not exist, enter none. Evaluate the limits: lim(x → -8) f(x), lim(x → 2^-) f(x), lim(x → 2^+) f(x), lim(x → ∞) f(x).

Solve the limit evaluation problem using a graph of the function f(x). Analyze the left-hand and right-hand limits at specific points and interpret the behavior as x approaches infinity. This problem explores key concepts in limits, continuity, and discontinuity, suitable for Grades 11-12 and introductory college calculus.

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