https://file.aiquickdraw.com/mathbot/file/17273717433590c49k7h3.jpg

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.

Use the graph of the function g(x) to state the values of the following limits:

1. \( \lim_{{x \to 2^-}} g(x) \)
2. \( \lim_{{x \to 2^+}} g(x) \)
3. \( \lim_{{x \to 2}} g(x) \)
4. \( \lim_{{x \to 5^-}} g(x) \)
5. \( \lim_{{x \to 5^+}} g(x) \)
6. \( \lim_{{x \to 5}} g(x) \).

Analyze the graph of the function g(x) and determine the left-hand and right-hand limits as x approaches 2 and 5. This includes determining whether the limits exist and understanding the concept of continuity. Suitable for students in Grades 11-12 studying calculus or advanced pre-calculus.

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