https://file.aiquickdraw.com/mathbot/file/17343104193730hv937ei.jpg

Let's solve the given limits step by step using the provided graph.

---

### (a) \(\lim_{x \to -2^-} f(x)\)

- This represents the **left-hand limit** as \(x\) approaches \(-2\).
- From the graph, observe the behavior of \(f(x)\) as \(x\) approaches \(-2\) from the left. The graph approaches \(y = 1\).
- **Answer**: \(\lim_{x \to -2^-} f(x) = 1\)

---

### (b) \(\lim_{x \to -2^+} f(x)\)

- This represents the **right-hand limit** as \(x\) approaches \(-2\).
- From the graph, observe the behavior of \(f(x)\) as \(x\) approaches \(-2\) from the right. The graph also approaches \(y = 1\).
- **Answer**: \(\lim_{x \to -2^+} f(x) = 1\)

---

### (c) \(\lim_{x \to 3} f(x)\)

- Here, \(x\) approaches \(3\) from both sides (two-sided limit).
- From the graph, as \(x\) approaches \(3\), the function \(f(x)\) does not approach a single value because there is a **discontinuity**: 
  - The graph shows an open circle at \(y = -1\) (indicating the function does not take that value).
  - There is a filled circle at \(y = 2\), which is the actual value of \(f(3)\).
- Since the left-hand limit and right-hand limit **do not match**, the two-sided limit does not exist.
- **Answer**: \(\lim_{x \to 3} f(x) \text{ does not exist.}\)

---

### Final Answers:
1. (a) \(\lim_{x \to -2^-} f(x) = 1\)  
2. (b) \(\lim_{x \to -2^+} f(x) = 1\)  
3. (c) \(\lim_{x \to 3} f(x) = \text{Does not exist}\)

---

Would you like me to clarify further or explore related examples?

### 5 Related Questions:
1. What are left-hand and right-hand limits? How are they defined?
2. How can you determine if a function is continuous at a point using limits?
3. How do open and closed circles on a graph affect limit evaluation?
4. What is the difference between limits and actual function values at a point?
5. How can we identify when a limit does not exist?

**Tip**: When evaluating limits graphically, carefully observe the behavior of the function as it approaches the point from both sides. If they match, the limit exists!

Use the graph provided to evaluate the limits:
(a) \( \lim_{x \to -2^-} f(x) \)
(b) \( \lim_{x \to -2^+} f(x) \)
(c) \( \lim_{x \to 3} f(x) \)

Learn how to evaluate limits using a graph. This example covers left-hand and right-hand limits, as well as cases where the limit does not exist due to discontinuity. Suitable for Grades 11-12 students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation