Let's analyze the function $f(x)$ from the graph provided and address the limit-related questions when $x = 3$.

Key Points from the Graph:

1. $\lim_{x \to 3^-} f(x)$:

   • This represents the left-hand limit as $x$ approaches 3 from the left. From the graph, as $x$ approaches 3 from the left, the function value approaches $-3$.
   • Answer: $\lim_{x \to 3^-} f(x) = -3$.

2. $\lim_{x \to 3^+} f(x)$:

   • This is the right-hand limit as $x$ approaches 3 from the right. As $x \to 3^+$, the graph also shows the function approaching $-3$.
   • Answer: $\lim_{x \to 3^+} f(x) = -3$.

3. $\lim_{x \to 3} f(x)$:

   • This is the general limit as $x$ approaches 3. Since both the left-hand and right-hand limits are equal (both are $-3$), the general limit exists and is $-3$.
   • Answer: $\lim_{x \to 3} f(x) = -3$.

4. $f(3)$:

   • This is the value of the function at $x = 3$. From the graph, there is a hole at $x = 3$, indicating that the function is not defined at this point.
   • Answer: $f(3)$ is undefined.

Final Answers:

1. $\lim_{x \to 3^-} f(x) = -3$
2. $\lim_{x \to 3^+} f(x) = -3$
3. $\lim_{x \to 3} f(x) = -3$
4. $f(3)$ is undefined.

Would you like more details on limits or have any other questions?

Here are some follow-up questions:

1. What is the difference between a one-sided and two-sided limit?
2. How can you determine if a function is continuous at a given point?
3. Why is $f(3)$ undefined in this graph?
4. What conditions are needed for the existence of $\lim_{x \to a} f(x)$?
5. Can you describe an example where the left and right limits are not equal?

Tip: When analyzing limits, always check for any discontinuities like holes, jumps, or vertical asymptotes in the graph.