The image shows a graph of a function $f(x)$ along with several limit expressions (labeled (I) to (VI)) on the top of the page. The graph provides values of the function $f(x)$ on a segment of the real line, with special behavior near $x = 0$.

Here's a breakdown of the function's behavior based on the graph:

1. Domain Analysis: The function $f(x)$ is defined for some parts of the interval around $x = 0$. It is defined on intervals such as $(-3, -2)$ and $(1, 3)$, while it has a discontinuity or undefined region near $x = 0$.

2. Behavior Near Zero: The function approaches different values as $x$ approaches $0$ from the left and right:

   • As $x \to 0^+$, $f(x)$ appears to approach infinity.
   • As $x \to 0^-$, $f(x)$ approaches negative infinity.

3. Values at Points:

   • At $x = -3$ and $x = 2$, there are open circles, suggesting these points are not included in the domain of $f(x)$ at those values.
   • At $x = -2$, there’s a closed circle, indicating $f(-2)$ is defined.

4. Limit Expressions: The top section of the image includes several limit expressions that refer to the behavior of $f(x)$ as $x$ approaches specific points. Here’s an interpretation:

   • (I) $\lim_{x \to -1^+} f(x)$
   • (II) $\lim_{x \to -1^-} f(x)$
   • (III) $\lim_{x \to 1^-} f(x)$
   • (IV) $\lim_{x \to 1^+} f(x)$
   • (V) $\lim_{x \to 3^-} f(x)$
   • (VI) $\lim_{x \to 3^+} f(x)$

To analyze these limits, you would observe the function's behavior on the graph near each point specified in the limit expressions.

Would you like help with evaluating these specific limits, or do you have questions about interpreting other parts of the graph?

Related Questions

1. What are the left and right-hand limits at $x = -1$ and $x = 1$?
2. How does the graph indicate whether $f(x)$ is continuous or discontinuous at certain points?
3. What is the value of $f(-2)$, and what does the graph suggest about this value?
4. How does the function behave as $x \to 3$?
5. What are the implications of the behavior as $x$ approaches zero for the overall continuity of $f(x)$?

Tip: When analyzing piecewise functions, always check both left-hand and right-hand limits to determine continuity at specific points.