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?

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.