To sketch the function $f(x)$ with the given properties, let's break down the key points:

1. Limit as $x \to -3$:

   • $\lim_{x \to -3} f(x) = 1$, meaning the function approaches 1 as $x$ gets close to -3.
   • However, $f(-3) = 3$, indicating a jump discontinuity at $x = -3$. The value of the function at $x = -3$ is not equal to the limit but is 3 instead.

2. Limits as $x \to 2$:

   • $\lim_{x \to 2^+} f(x) = 1$, meaning as $x$ approaches 2 from the right, the function approaches 1.
   • $\lim_{x \to 2^-} f(x) = -1$, meaning as $x$ approaches 2 from the left, the function approaches -1.
   • $f(2)$ is undefined, so there is a discontinuity (probably a removable or infinite discontinuity) at $x = 2$.

Visualizing the sketch:

• At $x = -3$, the function approaches $y = 1$, but there's a jump to $f(-3) = 3$, creating a point at (−3, 3) that stands above the limit line $y = 1$.
• At $x = 2$, there is a jump discontinuity because the function approaches 1 from the right and -1 from the left. The point $f(2)$ is undefined, leaving a gap at $x = 2$.
• For all other values of $x$, the function is continuous and behaves according to these limits.

A basic sketch would look like:

1. As $x \to -3$, the curve approaches $y = 1$ from both sides, but at $x = -3$, there is a dot at $y = 3$.
2. As $x \to 2$, the function approaches $y = -1$ from the left and $y = 1$ from the right, but there is no point at $x = 2$.

Let me know if you'd like a graph or further details!

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Would you like a visual graph, or any clarifications?

Here are 5 related questions to explore:

1. How do you distinguish between jump and removable discontinuities?
2. What are the implications of having an undefined point in a continuous function?
3. How does the behavior of one-sided limits influence the overall continuity of a function?
4. How can we describe continuity in terms of limits and function values?
5. What other types of discontinuities can occur in functions?

Tip: When sketching a function based on limits, focus on where the function approaches and what happens at critical points like jumps or undefined values.