Let's analyze the given properties step by step and sketch a graph of the function based on them:

1. $\lim_{x \to 3} f(x) = 1$ means that as $x$ approaches 3, the function approaches 1. The function has a limit of 1 at $x = 3$, but it doesn't necessarily equal 1 there.

2. $f(-3) = 3$ tells us that the function equals 3 at $x = -3$. This is a specific point on the graph: the coordinate $(-3, 3)$.

3. $\lim_{x \to 2^+} f(x) = 1$ implies that as $x$ approaches 2 from the right (i.e., values just greater than 2), the function approaches 1.

4. $\lim_{x \to 2^-} f(x) = -1$ implies that as $x$ approaches 2 from the left (i.e., values just less than 2), the function approaches -1.

5. $f(2)$ is undefined, meaning that the function has a discontinuity at $x = 2$.

Interpreting the behavior:

• At $x = 3$, the limit is 1, but the function could be different at $x = 3$, so we should put a hole at $x = 3$ with the function approaching 1.
• At $x = -3$, the function passes through the point $(-3, 3)$.
• At $x = 2$, the function is undefined, but we have different limits from both sides. From the left, it approaches -1, and from the right, it approaches 1. This is a jump discontinuity.

Based on these points, the graph would:

1. Pass through $(-3, 3)$.
2. Have a hole at $x = 3$ approaching $y = 1$.
3. Jump from $-1$ to $1$ at $x = 2$, with the function undefined at exactly $x = 2$.

Would you like me to generate a sketch of this graph?

Related questions:

1. What type of discontinuity is at $x = 2$, and why?
2. How does the graph behave near $x = 3$ and $x = 2$?
3. What would happen if the limit as $x$ approaches 3 was different from the value of $f(3)$?
4. How would the graph change if $f(2)$ were defined?
5. Can a function be continuous but still have undefined points like in this case?

Tip:

When dealing with limits, always pay attention to the direction of approach (left-hand or right-hand limits) to identify potential jump discontinuities.