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.