https://file.aiquickdraw.com/mathbot/file/1726092615963aebweurj.jpg

Let's analyze the graph and statements based on the image you uploaded.

### 1. **Statement A: \( \lim_{x \to -2} f(x) = -2 \)**
   - From the graph, as \( x \) approaches \(-2\), the function value approaches \(-2\). Therefore, this statement is **true**.

### 2. **Statement B: \( f(x) \text{ is continuous at } x = 0 \)**
   - Continuity at \( x = 0 \) requires that:
     1. \( f(0) \) is defined.
     2. \( \lim_{x \to 0} f(x) \) exists.
     3. \( \lim_{x \to 0} f(x) = f(0) \).
   - Looking at the graph, there is a jump discontinuity at \( x = 0 \) (since the left and right limits don't match), so this statement is **false**.

### 3. **Statement C: \( \lim_{x \to 2} f(x) = \infty \)**
   - At \( x = 2 \), the function approaches a finite value (which appears to be \( 2 \)), not \( \infty \). Therefore, this statement is **false**.

### 4. **Statement D: \( \lim_{x \to 4} f(x) = f(4) \)**
   - For \( x = 4 \), the graph shows that the limit as \( x \to 4 \) from the left is equal to \( f(4) \), which is defined and equals \( 4 \). So, this statement is **true**.

### 5. **Statement E: \( \lim_{x \to 0^+} f(x) = \lim_{x \to 0^-} f(x) \)**
   - As \( x \) approaches \( 0 \) from the left (\( x \to 0^- \)), the function value approaches 2. As \( x \) approaches 0 from the right (\( x \to 0^+ \)), the function value approaches 4. These limits are not equal, so this statement is **false**.

### Correct Answers:
- **A** and **D** are correct.

Would you like further details on any of these points?

### 5 Relative Questions:
1. What conditions must be met for a function to be continuous at a point?
2. How can we evaluate limits graphically at specific points?
3. What causes a discontinuity at \( x = 0 \) on the graph?
4. Why is \( \lim_{x \to 2} f(x) \) not infinite in this case?
5. How can we determine if left-hand and right-hand limits are equal?

### Tip:
Always check both the left-hand and right-hand limits when determining whether a function is continuous at a point or whether the limit exists.

Using the figure below, which of the following statements are true?

Analyze the graph and determine whether the limits and continuity conditions are satisfied. This problem covers key mathematical concepts, including limits, continuity, and the behavior of functions at specific points, such as x = 0 and x = 4. Suitable for students in Grades 11-12.

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