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.