From the graph provided, we are asked to evaluate the limit:

$\lim_{{x \to -4}} f(x)$

Steps to Analyze the Limit:

1. Check behavior from the left: As $x$ approaches -4 from the left ($x \to -4^-$), the function is increasing and seems to approach a value near $y = 4$.

2. Check behavior from the right: As $x$ approaches -4 from the right ($x \to -4^+$), the function is decreasing and also seems to approach the same value, $y = 4$.

Since the function approaches the same value of $y = 4$ from both the left and right sides of $x = -4$, the two-sided limit exists and is equal to 4.

Conclusion:

$\lim_{{x \to -4}} f(x) = 4$

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

Here are 5 related questions you might find interesting:

1. How do you determine if a limit exists based on a graph?
2. What is the difference between one-sided and two-sided limits?
3. How can limits help in determining continuity at a point?
4. What happens if the function approaches different values from the left and right of a point?
5. How can limits be used in the definition of derivatives?

Tip: If a function has different left-hand and right-hand limits, the overall limit does not exist at that point!