https://file.aiquickdraw.com/mathbot/file/1726871920132hek2zfls.jpg

Let's analyze the graph and the given statements step-by-step.

1. **\( f(2) \) is defined.**
   - This means that there is a defined value of \( f \) at \( x = 2 \). In the graph, at \( x = 2 \), there is a hole (open circle) which indicates that the function is not defined there. Hence, \( f(2) \) is **not defined**.

2. **\( \lim\limits_{x \to 2} f(x) \) exists.**
   - This refers to the limit of the function as \( x \) approaches 2 from both sides. From the graph, as \( x \) approaches 2 from the left and right, \( f(x) \) approaches the same value (looks like \( 1 \)). Hence, the limit \( \lim\limits_{x \to 2} f(x) \) **exists**.

3. **\( \lim\limits_{x \to 2} f(x) = f(2) \).**
   - For this to be true, both the limit as \( x \) approaches 2 and \( f(2) \) must exist and be equal. From the previous analysis, \( f(2) \) is not defined, so this statement is **false**.

4. **The function is continuous at \( x = 2 \).**
   - A function is continuous at \( x = 2 \) if \( f(2) \) is defined, \( \lim\limits_{x \to 2} f(x) \) exists, and \( \lim\limits_{x \to 2} f(x) = f(2) \). Since \( f(2) \) is not defined, the function is **not continuous** at \( x = 2 \).

5. **The function is not continuous at \( x = 2 \).**
   - Since \( f(2) \) is not defined, the function is indeed **not continuous** at \( x = 2 \).

### Conclusion:
The correct statements are:
- \( \lim\limits_{x \to 2} f(x) \) exists.
- The function is not continuous at \( x = 2 \).

Would you like more details or have any other questions? Here are some related questions you might be interested in:

1. What are the different types of discontinuities in a function?
2. How do we find the limit of a function analytically?
3. What is the definition of continuity for a function?
4. Can a function have a limit at a point where it is not defined?
5. What is the graphical interpretation of a limit?

**Tip:** Remember, a hole in the graph indicates that the function is not defined at that point, but it does not necessarily affect the existence of the limit.

Select all statements below about the function f(x) that you agree with, based on the provided graph.

Analyze the graph of f(x) to determine if the function is defined at x = 2, if the limit exists, and whether the function is continuous at that point. Key concepts include limits, continuity, and points of discontinuity.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation