The image you uploaded contains a problem related to limits of a piecewise function, along with a graph and several limit expressions to evaluate. I'll analyze each part of the problem to provide the solutions.

Analysis of the Graph and Limit Expressions:

The graph shows a piecewise function with various points of interest, specifically at $x = -4$, $x = -2$, $x = 0$, $x = 2$, and $x = 4$. The y-values and behavior at these points will help determine the limits.

Limits to Evaluate:

Given the limit expressions and the graph, let's evaluate each one:

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

   • As $x$ approaches $-4$ from both sides, the graph appears to approach the same value (y = 2).
   • Answer: $\lim_{x \to -4} f(x) = 2$

2. $\lim_{x \to -2} f(x)$

   • As $x$ approaches $-2$, the function seems to approach a specific value (y = 2) from both sides.
   • Answer: $\lim_{x \to -2} f(x) = 2$

3. $\lim_{x \to 0} f(x)$

   • As $x$ approaches 0 from both sides, the function does not converge to a single value (due to the jump discontinuity), hence the limit does not exist.
   • Answer: $\lim_{x \to 0} f(x)$ does not exist.

4. $\lim_{x \to 2^-} f(x)$

   • As $x$ approaches 2 from the left side, the function approaches y = 3.
   • Answer: $\lim_{x \to 2^-} f(x) = 3$

5. $\lim_{x \to 2^+} f(x)$

   • As $x$ approaches 2 from the right side, the function also approaches y = 3.
   • Answer: $\lim_{x \to 2^+} f(x) = 3$

6. $\lim_{x \to 4} f(x)$

   • As $x$ approaches 4 from both sides, the function seems to approach a single value (y = 1).
   • Answer: $\lim_{x \to 4} f(x) = 1$

Summary of Answers:

1. $\lim_{x \to -4} f(x) = 2$
2. $\lim_{x \to -2} f(x) = 2$
3. $\lim_{x \to 0} f(x)$ does not exist.
4. $\lim_{x \to 2^-} f(x) = 3$
5. $\lim_{x \to 2^+} f(x) = 3$
6. $\lim_{x \to 4} f(x) = 1$

Would you like further details or have any other questions?

Relative Questions:

1. What is the significance of one-sided limits in determining the behavior of a function?
2. How can you determine if a limit does not exist from a graph?
3. What are jump discontinuities, and how do they affect limits?
4. How do piecewise functions differ from continuous functions in terms of limit evaluation?
5. What role does symmetry play in evaluating limits of piecewise functions?

Tip:

When evaluating limits graphically, always check both sides of the point of interest to see if the function approaches the same value from the left and the right. If not, the limit may not exist at that point.