Let's analyze the graph and solve the limits based on the function $f(x)$.

1. $\lim\limits_{x \to -3} f(x)$: As $x \to -3$, the function approaches a value from both sides. From the left and the right, the graph approaches $f(x) = 4$. So, $\lim\limits_{x \to -3} f(x) = 4$

2. $\lim\limits_{x \to 3^+} f(x)$ (right-hand limit as $x \to 3$): From the right of $x = 3$, the graph approaches $f(x) = 5$, so $\lim\limits_{x \to 3^+} f(x) = 5$

3. $\lim\limits_{x \to 3} f(x)$: The left-hand limit of $f(x)$ as $x \to 3$ approaches $f(x) = 2$, but the right-hand limit approaches $f(x) = 5$. Since the left and right limits are not equal, this limit does not exist (DNE). $\lim\limits_{x \to 3} f(x) = \text{DNE}$

4. $\lim\limits_{x \to 2} f(x)$: As $x \to 2$, the graph approaches $f(x) = 3$ from both sides, so $\lim\limits_{x \to 2} f(x) = 3$

5. $\lim\limits_{x \to 2^+} f(x)$: As $x \to 2$ from the right, the graph again approaches $f(x) = 3$, so $\lim\limits_{x \to 2^+} f(x) = 3$

6. $\lim\limits_{x \to 4} f(x)$: As $x \to 4$, the graph approaches $f(x) = 4$ from both sides, so $\lim\limits_{x \to 4} f(x) = 4$

7. $\lim\limits_{x \to 4^+} f(x)$: From the right of $x = 4$, the graph approaches $f(x) = 4$, so $\lim\limits_{x \to 4^+} f(x) = 4$

Summary of Limits:

• $\lim\limits_{x \to -3} f(x) = 4$
• $\lim\limits_{x \to 3^+} f(x) = 5$
• $\lim\limits_{x \to 3} f(x) = \text{DNE}$
• $\lim\limits_{x \to 2} f(x) = 3$
• $\lim\limits_{x \to 2^+} f(x) = 3$
• $\lim\limits_{x \to 4} f(x) = 4$
• $\lim\limits_{x \to 4^+} f(x) = 4$

Let me know if you'd like any clarifications!

Related Questions:

1. What happens when a function has a jump discontinuity?
2. How are left-hand and right-hand limits different from general limits?
3. Can a limit exist if the function is undefined at that point?
4. How do we define limits at infinity?
5. What are some conditions for limits to not exist?

Tip: Always check the behavior of the function from both the left and right sides of a point to confirm if a limit exists.