Let's analyze the graph and the corresponding questions to determine the values or limits of the function $f(x)$.

1. Understanding the Graph:

• The graph shows a piecewise function with two different segments.
• The first segment (on the left) appears to be a curve that ends at $x = 1$ with a hole (open circle).
• The second segment (on the right) is a straight line starting at $x = 1$ with a filled dot, indicating the function is defined there.
• The x-axis ranges from about $-1$ to 4, and the y-axis from 0 to 5.

2. Analysis of Given Quantities:

a. $f(0)$:

• At $x = 0$, the graph intersects the y-axis at $y = 2$.
• Answer: $f(0) = 2$.

b. $\lim_{x \to 0} f(x)$:

• As $x$ approaches 0 from both sides, the function value approaches $y = 2$.
• Answer: $\lim_{x \to 0} f(x) = 2$.

c. $f(2)$:

• At $x = 2$, the graph has a filled circle at $y = 3$.
• Answer: $f(2) = 3$.

d. $\lim_{x \to 2} f(x)$:

• As $x$ approaches 2 from both sides, the function values approach $y = 3$.
• Answer: $\lim_{x \to 2} f(x) = 3$.

Summary:

• $f(0) = 2$
• $\lim_{x \to 0} f(x) = 2$
• $f(2) = 3$
• $\lim_{x \to 2} f(x) = 3$

These values indicate that the function is continuous at $x = 0$ and $x = 2$.

Would you like any further details or have additional questions?

Follow-up Questions:

1. What is the definition of a limit, and how is it evaluated graphically?
2. What does it mean when a graph has a hole (open circle) at a point?
3. How do you determine the continuity of a function from its graph?
4. What is the difference between a limit and the actual value of the function at a point?
5. How would you explain the concept of piecewise functions and their graphs?
6. What are the common types of discontinuities in a function?
7. How does one determine if a limit does not exist from a graph?
8. Can a function have a limit at a point where it is not defined?

Tip:

When analyzing limits graphically, always check the behavior of the function as it approaches the point from both the left and right sides.