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To solve the problem and determine \(\lim_{x \to -2} f(x)\) using the provided graph:

### Step-by-Step Analysis:
1. **Examine the behavior of the function near \(x = -2\)**: The key concept of limits is to analyze how the function behaves as \(x\) approaches a particular value, in this case, \(x = -2\).

2. **From the left-hand side (\(x \to -2^-\))**: As \(x\) approaches \(-2\) from the left (values slightly less than \(-2\)), the graph indicates that the function approaches \(y = 6\).

3. **From the right-hand side (\(x \to -2^+\))**: As \(x\) approaches \(-2\) from the right (values slightly greater than \(-2\)), the graph also approaches \(y = 6\).

4. **Conclusion on the limit**: Since both the left-hand limit and the right-hand limit exist and are equal, the overall limit as \(x\) approaches \(-2\) exists and is equal to 6.

Thus, the correct answer is:
\[
\lim_{x \to -2} f(x) = 6
\]

This corresponds to option **5** in the list provided.

Would you like further explanation on limits or related concepts? Here are some follow-up questions to deepen your understanding:

1. How do you differentiate between one-sided and two-sided limits?
2. What happens if the left-hand and right-hand limits are not equal?
3. How can you determine if a function is continuous at a given point?
4. What are the conditions for a limit to exist at a specific point?
5. How does the behavior of the graph around a vertical asymptote affect the limit?

**Tip**: When analyzing limits graphically, always check the behavior from both sides of the target point to ensure the function approaches the same value from each direction.

Use the graph to determine \(\lim_{x \to -2} f(x)\).

Analyze the graph of a function to determine the limit as \(x\) approaches -2. This problem covers key concepts such as limits, one-sided limits, and the graphical interpretation of limits. Suitable for students in Grades 10-12.

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