https://file.aiquickdraw.com/mathbot/file/1726440119605kypc9t20.jpg

Let's analyze the image and the questions asked in the problem.

We are working with the graph of a function \( f(x) \), and the question seems to involve finding limits and the value of the function at \( x = -6 \).

The tasks ask you to:
1. Determine the left-hand limit \( \lim_{x \to -6^-} f(x) \) from the graph.
2. Determine the right-hand limit \( \lim_{x \to -6^+} f(x) \) from the graph.
3. Determine whether \( \lim_{x \to -6} f(x) \) exists.
4. Find \( f(-6) \), the actual value of the function at \( x = -6 \), if it exists.
5. Make a conclusion about the continuity of \( f(x) \) at \( x = -6 \).

### Analyzing the graph at \( x = -6 \):
1. **Left-hand limit** \( \lim_{x \to -6^-} f(x) \): As \( x \) approaches -6 from the left, the function approaches **3**.
   
2. **Right-hand limit** \( \lim_{x \to -6^+} f(x) \): As \( x \) approaches -6 from the right, the function also approaches **3**.

3. **Overall limit** \( \lim_{x \to -6} f(x) \): Since both the left-hand and right-hand limits are equal, the limit at \( x = -6 \) exists and is equal to **3**.

4. **Function value** \( f(-6) \): According to the graph, at \( x = -6 \), there is a solid point at **5**, meaning \( f(-6) = 5 \).

5. **Conclusion**: The function \( f(x) \) is not continuous at \( x = -6 \) because \( \lim_{x \to -6} f(x) \neq f(-6) \).

### Filling in the answers:
- Left-hand limit: \( 3 \)
- Right-hand limit: \( 3 \)
- Limit: \( 3 \)
- \( f(-6) = 5 \)
- The function is **discontinuous** at \( x = -6 \) because \( \lim_{x \to -6} f(x) \neq f(-6) \).

Would you like more details on limits and continuity or have any questions?

### 5 Related Questions:
1. What is the general definition of a limit of a function?
2. How do you determine if a function is continuous at a specific point?
3. What is the difference between one-sided limits and two-sided limits?
4. Can a function have a limit but still be discontinuous at a point? Why?
5. What are some common types of discontinuities in functions?

### Tip:
When analyzing a graph for limits, always check for different behaviors on the left and right side of the point of interest.

Determine the left-hand and right-hand limits of f(x) at x = -6 and check the continuity of f(x) at that point.

Explore how to determine left-hand and right-hand limits at x = -6 for the function f(x) using its graph. Understand the conditions for continuity and why f(x) is discontinuous at this point.

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