https://file.aiquickdraw.com/mathbot/file/1725865029471okbvuxpr.jpg

Let's analyze the given graph of the function \( f \) and evaluate the truth of the three statements.

### 1. Statement I: 
\[
\lim_{{x \to 2}} f(x) = f(2)
\]

- Looking at the graph near \( x = 2 \), the value of \( f(2) \) is indicated by a **closed dot**, showing \( f(2) = 0 \).
- To check the limit \( \lim_{{x \to 2}} f(x) \), we observe the behavior of the function as \( x \) approaches 2 from both sides.
  - As \( x \to 2^- \), the graph approaches 0.
  - As \( x \to 2^+ \), the graph also approaches 0.
  - Therefore, \( \lim_{{x \to 2}} f(x) = 0 \).

Since both the limit and the function value at \( x = 2 \) are equal, **Statement I is true**.

---

### 2. Statement II:
\[
\lim_{{x \to 6^-}} f(x) = \lim_{{x \to 6^+}} f(x)
\]

- From the graph:
  - As \( x \to 6^- \), the function approaches the value \( f(x) = 3 \).
  - As \( x \to 6^+ \), the function approaches \( f(x) = -3 \).

Since the left-hand limit \( \lim_{{x \to 6^-}} f(x) = 3 \) and the right-hand limit \( \lim_{{x \to 6^+}} f(x) = -3 \) are not equal, **Statement II is false**.

---

### 3. Statement III:
\[
\lim_{{x \to 6}} f(x) = f(6)
\]

- From the graph:
  - \( f(6) \) is given by a **closed dot** at \( (6, 3) \), so \( f(6) = 3 \).
  - As analyzed above, \( \lim_{{x \to 6^-}} f(x) = 3 \) and \( \lim_{{x \to 6^+}} f(x) = -3 \), meaning that \( \lim_{{x \to 6}} f(x) \) does not exist.

Since the limit does not exist, **Statement III is false**.

---

### Conclusion:
- **Statement I is true.**
- **Statements II and III are false.**

Would you like more details or explanations?

Here are some related questions you can explore:
1. What is the formal definition of a limit?
2. How do left-hand and right-hand limits affect continuity?
3. What are some key conditions for a function to be continuous at a point?
4. What is the significance of open and closed dots in function graphs?
5. How can you identify jump discontinuities in a graph?

**Tip:** Always check the behavior of the function from both sides when calculating limits at a given point.

The figure above shows the graph of the function f. Which of the following statements are true?

I. lim_{x→2} f(x) = f(2)
II. lim_{x→6^-} f(x) = lim_{x→6^+} f(x)
III. lim_{x→6} f(x) = f(6)

Learn how to evaluate limits and continuity using a function graph. This problem involves checking if lim_{x→a} f(x) = f(a), as well as analyzing left-hand and right-hand limits. Suitable for Grades 11-12.

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