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Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If it does not, enter DNE.
a) f(1) b) lim x->1 f(x) c) f(4) d) lim x->4 f(x)

Let's analyze the graph to determine the values for each given quantity:

### a) \( f(1) \)
- **From the graph:** At \( x = 1 \), the function \( f(x) \) is the value of \( y \) corresponding to \( x = 1 \).
- **Observation:** There is a solid dot at \( x = 1 \) on the graph, which indicates that the function is defined at this point.
- **Answer:** \( f(1) = 4 \).

### b) \( \lim_{x \to 1} f(x) \)
- **From the graph:** The limit as \( x \) approaches 1 is the value that \( f(x) \) approaches as \( x \) gets closer to 1 from both the left and the right sides.
- **Observation:** Both the left-hand and right-hand limits as \( x \) approaches 1 are approaching the value \( y = 6 \).
- **Answer:** \( \lim_{x \to 1} f(x) = 6 \).

### c) \( f(4) \)
- **From the graph:** At \( x = 4 \), we need to find the corresponding \( y \)-value.
- **Observation:** There is no dot or solid point at \( x = 4 \), meaning the function is not defined at this point.
- **Answer:** \( f(4) = \text{DNE (Does Not Exist)} \).

### d) \( \lim_{x \to 4} f(x) \)
- **From the graph:** The limit as \( x \) approaches 4 is the value that \( f(x) \) approaches as \( x \) gets closer to 4 from both the left and the right sides.
- **Observation:** Both the left-hand and right-hand limits as \( x \) approaches 4 are approaching the value \( y = 8 \).
- **Answer:** \( \lim_{x \to 4} f(x) = 8 \).

### Summary of Answers:
- \( \text{a) } f(1) = 4 \)
- \( \text{b) } \lim_{x \to 1} f(x) = 6 \)
- \( \text{c) } f(4) = \text{DNE} \)
- \( \text{d) } \lim_{x \to 4} f(x) = 8 \)

Would you like further details or have any questions?

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Here are some follow-up questions that you might find helpful:

1. What is the significance of a limit in calculus?
2. How do left-hand and right-hand limits determine the existence of a limit at a point?
3. What conditions must be met for a function to be continuous at a point?
4. What role does the existence of a point in a graph play in determining function values?
5. Can a function have a limit at a point where it is not defined? Explain.

**Tip:** Understanding the concept of limits is crucial in calculus, especially when dealing with discontinuities in functions.

Explore how to evaluate function values and limits using the graph of function f. Learn about specific evaluations such as f(1), lim x->1 f(x), f(4), and lim x->4 f(x) with detailed explanations and observations.

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