Let's analyze the graph and the associated questions.

Graph Details:

• The graph shows the function $y = f(x)$ plotted on the coordinate plane.
• There are several key points where the function changes direction, has jumps, or is undefined.

Questions:

1. $f(2)$: This asks for the value of the function at $x = 2$.

   • At $x = 2$, the corresponding $y$-value is 4.
   • Answer: $f(2) = 4$.

2. $\lim_{x \to 2^-} f(x)$: This asks for the left-hand limit as $x$ approaches 2.

   • As $x$ approaches 2 from the left, the $y$-value approaches 1.
   • Answer: $\lim_{x \to 2^-} f(x) = 1$.

3. $\lim_{x \to 2^+} f(x)$: This asks for the right-hand limit as $x$ approaches 2.

   • As $x$ approaches 2 from the right, the $y$-value approaches 4.
   • Answer: $\lim_{x \to 2^+} f(x) = 4$.

4. $\lim_{x \to 2} f(x)$: This asks for the limit of $f(x)$ as $x$ approaches 2.

   • Since the left-hand limit ($1$) does not equal the right-hand limit ($4$), the overall limit does not exist.
   • Answer: $\lim_{x \to 2} f(x)$ does not exist.

Summary of Answers:

• $f(2) = 4$
• $\lim_{x \to 2^-} f(x) = 1$
• $\lim_{x \to 2^+} f(x) = 4$
• $\lim_{x \to 2} f(x)$ does not exist

Would you like further details on these concepts, or do you have any additional questions?

Here are five relative questions that build upon this information:

1. What is the value of $\lim_{x \to 4} f(x)$ based on the graph?
2. How would you determine $f(5)$ using the graph?
3. What is the significance of a jump discontinuity at $x = 2$ in the context of limits?
4. How does the right-hand limit at $x = 7$ compare to the left-hand limit at the same point?
5. If $\lim_{x \to 3} f(x) = 2$, what does this tell us about the behavior of $f(x)$ around $x = 3$?

Tip: When evaluating limits graphically, always compare the left-hand and right-hand limits to determine if the overall limit exists.