https://file.aiquickdraw.com/mathbot/file/17288433120600mqg66si.jpg

Let’s walk through the graph to evaluate the limits based on the function \(y = f(x)\).

The task asks for:
1. \( \lim_{x \to 1} f(x) \)  
2. \( \lim_{x \to 0} f(x) \)  
3. \( \lim_{x \to 4^-} f(x) \)

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### **Step-by-Step Evaluation**

1. **Evaluating \( \lim_{x \to 1} f(x) \):**
   - As \(x\) approaches 1 from both sides, the function values appear to get closer to **5** (where the curve meets the point at \( (1, 5) \)).
   - Thus, \( \lim_{x \to 1} f(x) = 5 \).

2. **Evaluating \( \lim_{x \to 0} f(x) \):**
   - As \(x\) approaches 0 from both the left and right, the function value stabilizes around **3** (since the graph passes smoothly through \((0, 3)\)).
   - So, \( \lim_{x \to 0} f(x) = 3 \).

3. **Evaluating \( \lim_{x \to 4^-} f(x) \):**
   - When \(x\) approaches 4 from the **left side**, the function value seems to approach **-3**.
   - There is no discontinuity as we approach 4 from the left.
   - Therefore, \( \lim_{x \to 4^-} f(x) = -3 \).

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### **Answer Matching**
Based on our evaluations:

- \( \lim_{x \to 1} f(x) = 5 \)
- \( \lim_{x \to 0} f(x) = 3 \)
- \( \lim_{x \to 4^-} f(x) = -3 \)

Thus, the correct answer is the option:  
- **\( \lim_{x \to 1} f(x) = 5 \), \( \lim_{x \to 0} f(x) = 3 \), \( \lim_{x \to 4^-} f(x) = -3 \)**

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### Follow-Up Questions:
1. What is the definition of a left-hand and right-hand limit?
2. How can you identify discontinuities in a graph based on the limit behavior?
3. What does it mean if a limit does not exist at a point?
4. How do piecewise functions affect limit evaluations?
5. What role do removable discontinuities play in limits?

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**Tip:** Always compare the left-hand and right-hand limits at a point. If they differ, the limit at that point does not exist!

Consider the graph of some function y = f(x). Evaluate the following limits: lim x→1 f(x), lim x→0 f(x), and lim x→4^- f(x).

In this problem, learn how to evaluate limits from a graph of y = f(x). We explore the limits as x approaches 1, 0, and 4 from both sides of the function, providing a detailed explanation of the function’s behavior.

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