To solve the given limits using the graph of the function $g(x)$, let's analyze each part:

1. $\lim\limits_{x \to -2} g(x)$

   • As $x$ approaches $-2$ from both the left and right sides, observe the behavior of the graph near $x = -2$.
   • From the graph, it seems that as $x$ approaches $-2$, the function approaches $g(x) = 1$.
   • Answer: $\lim\limits_{x \to -2} g(x) = 1$

2. $\lim\limits_{x \to 2} g(x)$

   • Here, examine how the function behaves as $x$ approaches 2 from the left and right sides.
   • The graph shows a discontinuity at $x = 2$, with the left-hand limit approaching $0$ and the right-hand limit approaching $4$. Since the left-hand and right-hand limits are not equal, the limit does not exist.
   • Answer: $\lim\limits_{x \to 2} g(x)$ does not exist.

3. $\lim\limits_{x \to 3} g(x)$

   • As $x$ approaches $3$, check how $g(x)$ behaves.
   • The graph indicates that as $x$ approaches $3$ from either side, $g(x)$ approaches $2$.
   • Answer: $\lim\limits_{x \to 3} g(x) = 2$

4. $\lim\limits_{x \to 0} g(x)$

   • Examine the graph near $x = 0$.
   • The function appears to smoothly approach $g(x) = 0$ as $x$ approaches $0$.
   • Answer: $\lim\limits_{x \to 0} g(x) = 0$

5. $g(2)$

   • Check the value of the function at $x = 2$ directly from the graph.
   • There is a solid dot at $x = 2$ and $g(x) = 3$, indicating the value of the function at this point.
   • Answer: $g(2) = 3$

Let me know if you need further details or explanations!

Relative Questions:

1. What is the difference between a one-sided limit and a two-sided limit?
2. How do you determine the continuity of a function at a given point?
3. What are the conditions under which a limit does not exist?
4. How can you identify if a function has a jump discontinuity from a graph?
5. What is the importance of the epsilon-delta definition in understanding limits?

Tip: Always check both left-hand and right-hand limits to ensure the overall limit exists when analyzing the behavior of functions at specific points.