https://file.aiquickdraw.com/mathbot/file/1726087154151eiuj7gve.jpg

Let's break this down step by step by analyzing the graph and answering the given questions.

### Part 1: \( f(1) \)
- From the graph, observe what happens at \( x = 1 \).
- At \( x = 1 \), there is a **hole** in the graph, indicating that \( f(1) \) is **undefined** (meaning it does not exist at \( x = 1 \)).
- Therefore, \( f(1) = \text{DNE} \) (Does Not Exist).

### Part 2: \( \lim_{x \to 1} f(x) \)
- The limit as \( x \to 1 \) looks at the behavior of the function as \( x \) approaches 1 from both sides (left and right).
- As \( x \) approaches 1 from the left, the graph approaches a value of approximately **0**.
- As \( x \) approaches 1 from the right, the graph also approaches the value **0**.
- Since both the left-hand and right-hand limits agree, the limit exists and equals **0**.

Thus, we have the following answers:
1. \( f(1) = \text{DNE} \)
2. \( \lim_{x \to 1} f(x) = 0 \)

Would you like more details on any part of this? Here are 5 related questions to further your understanding:

1. What is the significance of a hole in a graph at a point like \( x = 1 \)?
2. How can you determine if a limit exists at a point where the function itself is undefined?
3. What is the difference between a function value and a limit at the same point?
4. How can you find the limit at other points on the graph?
5. What role does continuity play in determining function values and limits?

**Tip:** When evaluating limits, always check the behavior of the function from both sides of the point of interest (left-hand limit and right-hand limit).

Use the graph of y = f(x) to compute the values of f(1) and the limit as x approaches 1 of f(x).

Solve the graph-based problem by determining f(1) and the limit as x approaches 1 using limit and continuity theorems. Ideal for high school students studying calculus.

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