Let's analyze the graph to determine the values for each given quantity:

a) $f(1)$

• From the graph: At $x = 1$, the function $f(x)$ is the value of $y$ corresponding to $x = 1$.
• Observation: There is a solid dot at $x = 1$ on the graph, which indicates that the function is defined at this point.
• Answer: $f(1) = 4$.

b) $\lim_{x \to 1} f(x)$

• From the graph: The limit as $x$ approaches 1 is the value that $f(x)$ approaches as $x$ gets closer to 1 from both the left and the right sides.
• Observation: Both the left-hand and right-hand limits as $x$ approaches 1 are approaching the value $y = 6$.
• Answer: $\lim_{x \to 1} f(x) = 6$.

c) $f(4)$

• From the graph: At $x = 4$, we need to find the corresponding $y$-value.
• Observation: There is no dot or solid point at $x = 4$, meaning the function is not defined at this point.
• Answer: $f(4) = \text{DNE (Does Not Exist)}$.

d) $\lim_{x \to 4} f(x)$

• From the graph: The limit as $x$ approaches 4 is the value that $f(x)$ approaches as $x$ gets closer to 4 from both the left and the right sides.
• Observation: Both the left-hand and right-hand limits as $x$ approaches 4 are approaching the value $y = 8$.
• Answer: $\lim_{x \to 4} f(x) = 8$.

Summary of Answers:

• $\text{a) } f(1) = 4$
• $\text{b) } \lim_{x \to 1} f(x) = 6$
• $\text{c) } f(4) = \text{DNE}$
• $\text{d) } \lim_{x \to 4} f(x) = 8$

Would you like further details or have any questions?

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Here are some follow-up questions that you might find helpful:

1. What is the significance of a limit in calculus?
2. How do left-hand and right-hand limits determine the existence of a limit at a point?
3. What conditions must be met for a function to be continuous at a point?
4. What role does the existence of a point in a graph play in determining function values?
5. Can a function have a limit at a point where it is not defined? Explain.

Tip: Understanding the concept of limits is crucial in calculus, especially when dealing with discontinuities in functions.