https://file.aiquickdraw.com/mathbot/file/17366911377834ioi0gbm.jpg

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

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### Given Problem
We are asked to:

1. **Evaluate \( f(1) \).**
2. **Evaluate \( \lim_{x \to 3} f(x) \).**
3. **Evaluate \( \lim_{x \to 4^+} f(x) \).**
4. **Evaluate \( \lim_{x \to 8} f(x) \).**

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### Step 1: \( f(1) \)
From the graph, at \( x = 1 \), the function \( f(x) \) is at \( y = 4 \).

Thus, \( f(1) = 4 \).

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### Step 2: \( \lim_{x \to 3} f(x) \)
To find \( \lim_{x \to 3} f(x) \), we examine the behavior of \( f(x) \) as \( x \) approaches 3 from both sides:

- **As \( x \to 3^- \):** The function approaches \( y = 2 \) (from the left side).
- **As \( x \to 3^+ \):** The function approaches \( y = 4 \) (from the right side).

Since the left-hand limit (\( 2 \)) and right-hand limit (\( 4 \)) are not equal, the limit **does not exist** (DNE).

Thus, \( \lim_{x \to 3} f(x) = \text{DNE} \).

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### Step 3: \( \lim_{x \to 4^+} f(x) \)
To find \( \lim_{x \to 4^+} f(x) \), we examine the behavior of \( f(x) \) as \( x \) approaches 4 from the right side:

- **As \( x \to 4^+ \):** The function immediately approaches \( y = 3 \).

Thus, \( \lim_{x \to 4^+} f(x) = 3 \).

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### Step 4: \( \lim_{x \to 8} f(x) \)
To find \( \lim_{x \to 8} f(x) \), we examine the behavior of \( f(x) \) as \( x \) approaches 8 from both sides:

- **As \( x \to 8^- \):** The function approaches \( y = 2 \) (from the left side).
- **As \( x \to 8^+ \):** The function approaches \( y = 2 \) (from the right side).

Since both sides agree, the limit exists and is \( 2 \).

Thus, \( \lim_{x \to 8} f(x) = 2 \).

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### Final Answers:
1. \( f(1) = 4 \)
2. \( \lim_{x \to 3} f(x) = \text{DNE} \)
3. \( \lim_{x \to 4^+} f(x) = 3 \)
4. \( \lim_{x \to 8} f(x) = 2 \)

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Would you like a detailed explanation of any specific part? Let me know!  

### Related Questions:
1. How do we determine continuity of a function at a point from a graph?
2. What are the differences between left-hand and right-hand limits?
3. Why does a limit fail to exist at certain points?
4. How does the graph of a piecewise function relate to limits and continuity?
5. Can a function have a limit even if it is undefined at a point?  

### Tip:
When evaluating limits from a graph, always check the left-hand and right-hand behaviors carefully, especially at points of discontinuity!

Enter each answer as a whole number (like -4, 0, or 253) or DNE for undefined or Does Not Exist based on the provided graph of the function y = f(x). Evaluate the following:
1. f(1)
2. lim x->3 f(x)
3. lim x->4+ f(x)
4. lim x->8 f(x)

Analyze the graph of y = f(x) to evaluate f(1), determine the existence of limits such as lim x->3 f(x), and solve problems involving left-hand and right-hand limits. Suitable for Grades 10-12 students.

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