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To solve the given problems using the graph of the function \( f \), let's analyze each part one by one:

### Given Parts:
1. \( f(1) \)
2. \( \lim_{{x \to 1}} f(x) \)
3. \( f(6) \)
4. \( \lim_{{x \to 6}} f(x) \)

### Analyzing the Graph:

Let's examine the graph:

1. **Point (a): \( f(1) \)**
   - To find \( f(1) \), we look for the y-coordinate of the function at \( x = 1 \). 
   - From the graph, there is an open circle at \( x = 1 \) with \( y = 3 \), meaning \( f(1) \) is undefined at that point.
   - **Answer: DNE (Does Not Exist).**

2. **Point (b): \( \lim_{{x \to 1}} f(x) \)**
   - To find the limit as \( x \to 1 \), we check the values of the function as \( x \) approaches 1 from both the left and right.
   - As \( x \to 1 \) from both sides, the function approaches \( y = 3 \).
   - **Answer: 3.**

3. **Point (c): \( f(6) \)**
   - To find \( f(6) \), we look for the y-coordinate of the function at \( x = 6 \).
   - The graph shows a point at \( x = 6 \) with \( y = 4 \) (a filled circle).
   - **Answer: 4.**

4. **Point (d): \( \lim_{{x \to 6}} f(x) \)**
   - To find the limit as \( x \to 6 \), we check the values of the function as \( x \) approaches 6 from both sides.
   - As \( x \to 6 \), the function approaches \( y = 2 \) from both sides (based on the continuous line before reaching \( x = 6 \)).
   - **Answer: 2.**

### Final Answers:

(a) \( f(1) = \text{DNE} \)  
(b) \( \lim_{{x \to 1}} f(x) = 3 \)  
(c) \( f(6) = 4 \)  
(d) \( \lim_{{x \to 6}} f(x) = 2 \)

Would you like more details on any part of the solution, or have any further questions?

### Related Questions:

1. What is the difference between a limit and the value of a function at a point?
2. How can discontinuities in a function be classified?
3. What is a removable discontinuity, and how does it appear on a graph?
4. What are the left-hand limit and right-hand limit, and how do they relate to the overall limit?
5. How can you determine if a function is continuous at a certain point?

**Tip:** When analyzing limits from a graph, always check for open and closed circles to understand where the function value is defined or undefined.

Use the graph of the function f to decide whether each quantity exists. (If an answer does not exist, enter DNE.) Find the following: (a) f(1), (b) lim (x->1) f(x), (c) f(6), (d) lim (x->6) f(x).

Learn how to evaluate limits and function values from graphs. This includes determining whether a function is defined at specific points and analyzing limits as x approaches certain values. This problem covers the key concepts of limits, continuity, and function evaluation, suitable for high school calculus students.

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