graph limx→0−f(x)=−1,limx→0+f(x)=2,f(0)=1  

To graph the function based on the given limits and the value of \( f(0) \), let's interpret the information:

1. **Left-hand limit**: \( \lim_{x \to 0^-} f(x) = -1 \)
   - As \( x \) approaches 0 from the left (negative side), the function approaches \( -1 \).

2. **Right-hand limit**: \( \lim_{x \to 0^+} f(x) = 2 \)
   - As \( x \) approaches 0 from the right (positive side), the function approaches \( 2 \).

3. **Function at \( x = 0 \)**: \( f(0) = 1 \)
   - The actual value of the function at \( x = 0 \) is \( 1 \), which differs from the left-hand and right-hand limits.

### Key features for the graph:

- **Discontinuity at \( x = 0 \)**: The left-hand and right-hand limits differ, so the function has a jump discontinuity at \( x = 0 \).
- **Behavior near \( x = 0 \)**:
  - As \( x \) approaches 0 from the left, \( f(x) \) gets closer to \( -1 \).
  - As \( x \) approaches 0 from the right, \( f(x) \) gets closer to \( 2 \).
- **At \( x = 0 \)**, the function jumps to \( f(0) = 1 \).

This would result in a graph where:
- The function gets closer to \( -1 \) from the left but never reaches 0.
- The function gets closer to \( 2 \) from the right but jumps to \( 1 \) at \( x = 0 \).
  
Would you like me to create a visual graph of this behavior?

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### Related Questions:
1. What is a jump discontinuity, and how is it different from a removable discontinuity?
2. Can a function be continuous at \( x = 0 \) if its left and right limits are different?
3. How can you find the limits from a graph of a function?
4. How do limits relate to the continuity of a function?
5. What are the different types of discontinuities in functions?

**Tip:** When graphing, always pay close attention to limit behaviors near critical points (like 0 here), as they can reveal key discontinuities.

graph limx→0−f(x)=−1,limx→0+f(x)=2,f(0)=1

This example explains how to graph a function based on limits from the left and right, where the left-hand limit is -1, the right-hand limit is 2, and the function value at x=0 is 1. Learn how to identify a jump discontinuity and analyze limit behavior. Suitable for advanced high school students or college-level mathematics.

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