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In this graph, we are asked to evaluate the limits of the function \( f(x) \) at specific points: \( x = -6 \), \( x = -2 \), and \( x = 4 \). Based on the visual features of the graph, we can analyze the behavior of \( f(x) \) as \( x \) approaches these values.

### 1. \( \lim\limits_{x \to -6} f(x) \)
- At \( x = -6 \), there is a solid point on the curve that indicates that the function is continuous and defined there. The value of \( f(-6) \) appears to be approximately 5.
- Therefore, \( \lim\limits_{x \to -6} f(x) = 5 \).

### 2. \( \lim\limits_{x \to -2} f(x) \)
- At \( x = -2 \), there is an open circle, meaning the function is not defined exactly at this point. However, the graph approaches a specific value from both sides.
- As \( x \) approaches \( -2 \) from both directions, the function approaches \( f(x) = 5 \), even though there is an open point.
- Therefore, \( \lim\limits_{x \to -2} f(x) = 5 \).

### 3. \( \lim\limits_{x \to 4} f(x) \)
- At \( x = 4 \), the function again has an open circle, so the function is not defined at exactly \( x = 4 \). However, looking at the behavior from both sides, the graph approaches \( f(x) = -5 \).
- Therefore, \( \lim\limits_{x \to 4} f(x) = -5 \).

### Summary of Limits:
- \( \lim\limits_{x \to -6} f(x) = 5 \)
- \( \lim\limits_{x \to -2} f(x) = 5 \)
- \( \lim\limits_{x \to 4} f(x) = -5 \)

Would you like further details or have any questions about this analysis?

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Here are five related questions to further explore:
1. What is the definition of a limit, and how is it applied to analyze a graph?
2. How do open and closed points affect the calculation of limits?
3. Can you always find limits for points where a function is undefined?
4. What are one-sided limits, and how would they differ from these results?
5. How does continuity at a point relate to the value of the limit?

**Tip:** Always check the behavior of the graph from both sides when determining limits at points with open or undefined values.

Evaluate \( \lim_{x \to -6} f(x) \), \( \lim_{x \to -2} f(x) \), and \( \lim_{x \to 4} f(x) \) based on the given graph.

Learn how to evaluate limits from a graph with this example focusing on the function f(x). We analyze the behavior of f(x) as x approaches specific points, addressing cases of continuity and open circles. Suitable for students in Grades 11-12.

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