https://file.aiquickdraw.com/mathbot/file/17264889501792op8k4mt.jpg

Let's analyze the graph and solve the given questions:

### 1. \( f(1) \):
We need to find the value of the function \( f(x) \) at \( x = 1 \). 

Looking at the graph, there is a filled dot at \( (1, 2) \), indicating that the value of \( f(1) = 2 \). So:
\[
f(1) = 2
\]

### 2. \( \lim_{x \to 1} f(x) \):
Now we calculate the limit of \( f(x) \) as \( x \) approaches 1. This requires checking both the left-hand limit (\( x \to 1^- \)) and the right-hand limit (\( x \to 1^+ \)).

- As \( x \to 1^- \), the graph approaches 3 (from the left).
- As \( x \to 1^+ \), the graph approaches 3 (from the right).

Since both limits agree, we conclude:
\[
\lim_{x \to 1} f(x) = 3
\]

### Summary:
- \( f(1) = 2 \)
- \( \lim_{x \to 1} f(x) = 3 \)

Would you like further details or clarifications?

Here are 5 related questions to further explore these ideas:
1. What happens if the limit from the left and right do not match?
2. Can a function be discontinuous at a point and still have a limit at that point?
3. How do you determine if a function is continuous at a certain point?
4. What are the conditions for the existence of a limit?
5. How do limits apply to real-world situations like speed or temperature changes?

**Tip:** Always check both one-sided limits when evaluating limits to ensure they match, confirming the existence of the overall limit.

Use the graph of the function below to evaluate f(1) and the limit of f(x) as x approaches 1.

This problem demonstrates how to evaluate f(1) and the limit of f(x) as x approaches 1 using a graph. Key concepts include limits, continuity, and evaluating function values from a graph. Suitable for students in Grades 11-12.

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