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.