Let's analyze the limits of the function $g(x)$ using the graph provided. Here are the values of each limit:

1. $\lim_{{x \to 2^-}} g(x)$:

   • As $x$ approaches 2 from the left, the values of $g(x)$ approach 3.
   • So, $\lim_{{x \to 2^-}} g(x) = 3$.

2. $\lim_{{x \to 2^+}} g(x)$:

   • As $x$ approaches 2 from the right, the values of $g(x)$ approach 1.
   • So, $\lim_{{x \to 2^+}} g(x) = 1$.

3. $\lim_{{x \to 2}} g(x)$:

   • Since the left-hand limit ($3$) and the right-hand limit ($1$) are different, the limit does not exist at $x = 2$.
   • Therefore, $\lim_{{x \to 2}} g(x)$ does not exist.

4. $\lim_{{x \to 5^-}} g(x)$:

   • As $x$ approaches 5 from the left, $g(x)$ approaches 2.
   • So, $\lim_{{x \to 5^-}} g(x) = 2$.

5. $\lim_{{x \to 5^+}} g(x)$:

   • As $x$ approaches 5 from the right, $g(x)$ also approaches 2.
   • So, $\lim_{{x \to 5^+}} g(x) = 2$.

6. $\lim_{{x \to 5}} g(x)$:

   • Since both the left-hand limit and right-hand limit are equal to 2, the limit exists and is equal to 2.
   • Therefore, $\lim_{{x \to 5}} g(x) = 2$.

Would you like more details on any of these limits, or do you have other questions?

Here are 5 related questions:

1. How do we determine if a limit exists at a point?
2. What is the significance of the left-hand and right-hand limits?
3. How does a jump discontinuity affect the existence of a limit?
4. Can a function be continuous at a point if its limit does not exist there?
5. What are the different types of discontinuities in a graph?

Tip: To find the limit of a function graphically, always check the behavior of the function from both sides of the point of interest.