https://file.aiquickdraw.com/mathbot/file/17270545146022w0zjtrh.jpg

Let’s analyze the graph to solve each part of the problem.

### 1. \( f(1) \)
At \( x = 1 \), the graph shows that the filled dot (indicating the value of the function) is located at \( y = 2 \). Thus, \( f(1) = 2 \).

### 2. \( \lim_{x \to 0} f(x) \)
As \( x \) approaches 0 from both sides, we observe that the graph approaches the point \( y = 2 \). Thus, \( \lim_{x \to 0} f(x) = 2 \).

### 3. \( \lim_{x \to 5^+} f(x) \)
This is the limit as \( x \) approaches 5 from the right. As we approach \( x = 5 \) from values greater than 5, the graph appears to approach \( y = 3 \). Thus, \( \lim_{x \to 5^+} f(x) = 3 \).

### 4. \( \lim_{x \to 5^-} f(x) \)
This is the limit as \( x \) approaches 5 from the left. As we approach \( x = 5 \) from values less than 5, the graph approaches \( y = 1 \). Thus, \( \lim_{x \to 5^-} f(x) = 1 \).

### Summary of Answers:
- \( f(1) = 2 \)
- \( \lim_{x \to 0} f(x) = 2 \)
- \( \lim_{x \to 5^+} f(x) = 3 \)
- \( \lim_{x \to 5^-} f(x) = 1 \)

Would you like any further details or clarifications on these solutions?

Here are 5 related questions to explore:
1. What is the value of \( f(5) \) based on the graph?
2. Does the limit \( \lim_{x \to 9} f(x) \) exist?
3. What can you infer if \( \lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x) \)?
4. How would you define continuity at a point based on this graph?
5. How can we use limits to describe behavior at discontinuities in the function?

**Tip:** When calculating limits, always look carefully at the graph’s behavior from both the left and right to check for any discrepancies or jumps at a point.

Given a graph, calculate f(1), lim(x→0)f(x), lim(x→5+)f(x), and lim(x→5-)f(x).

Learn how to find f(1), and calculate limits approaching x from both sides (lim(x→0), lim(x→5+), lim(x→5-)) using a piecewise function graph. Suitable for high school students, this problem covers limits, continuity, and function analysis.

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