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To solve this problem, we need to determine the limit of the function \( f(x) \) as \( x \) approaches 4, using the graph provided.

### Step-by-step Solution:

1. **Identify the left-hand limit as \( x \to 4^- \):**
   - As \( x \) approaches 4 from the left, the values of \( f(x) \) approach \( -2 \).

2. **Identify the right-hand limit as \( x \to 4^+ \):**
   - As \( x \) approaches 4 from the right, the values of \( f(x) \) approach 2.

3. **Check if the left-hand limit equals the right-hand limit:**
   - The left-hand limit is \( -2 \) and the right-hand limit is \( 2 \).
   - Since \( -2 \neq 2 \), the left-hand limit and right-hand limit are not equal.

4. **Conclusion:**
   - The limit does not exist because the left-hand and right-hand limits are not equal.

### Answer:
The correct answer is: **"does not exist because the left- and right-hand limits are not equal."**

Would you like more details on limits or have any further questions?

### 5 Related Questions:
1. What is the definition of a limit in calculus?
2. How do you determine the left-hand and right-hand limits from a graph?
3. What does it mean for a function to be continuous at a point?
4. How do limits relate to the concept of derivatives?
5. What is the difference between one-sided limits and two-sided limits?

### Tip:
When analyzing limits on a graph, always check both the left-hand and right-hand limits to see if they converge to the same value. If they do, the limit exists; if they don't, the limit does not exist.

Use the graph of f shown to find the indicated limit or function value. If the limit does not exist, explain why.

lim(x→4)f(x) = ?

Explore how to determine the limit of the function f(x) as x approaches 4 using a graph. This example demonstrates the importance of checking both left-hand and right-hand limits to determine if a limit exists. Perfect for high school students studying calculus concepts like limits and continuity.

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