To address the questions from the graph in Figure 14, we need to analyze the function $f(x)$ at $x = 4$ and its behavior around this point.

Estimation from the graph:

1. Left-hand limit $\lim_{x \to 4^-} f(x)$: As $x$ approaches 4 from the left, the graph seems to approach the value $y = 2$. So, $\lim_{x \to 4^-} f(x) = 2$

2. Right-hand limit $\lim_{x \to 4^+} f(x)$: As $x$ approaches 4 from the right, the graph also appears to approach $y = 2$. Thus, $\lim_{x \to 4^+} f(x) = 2$

3. Two-sided limit $\lim_{x \to 4} f(x)$: Since both the left-hand and right-hand limits are equal to 2, the two-sided limit exists and is: $\lim_{x \to 4} f(x) = 2$

4. Value of $f(4)$: From the graph, there is a solid point at $x = 4$, with $y = 3$, indicating that the function's value at $x = 4$ is: $f(4) = 3$

Final answers:

• $\lim_{x \to 4^-} f(x) = 2$
• $\lim_{x \to 4^+} f(x) = 2$
• $\lim_{x \to 4} f(x) = 2$
• $f(4) = 3$

Would you like further details or explanations?

Here are 5 related questions to deepen your understanding:

1. What happens if the left-hand and right-hand limits are not equal?
2. How does the graph's continuity relate to the limits and function value at $x = 4$?
3. What is the difference between a limit and the actual value of the function?
4. How do you calculate limits when a graph is not available?
5. Can a function have a limit at a point even if the function is not defined there?

Tip: A function is continuous at a point $x = a$ if the left-hand limit, right-hand limit, and the function's value at $a$ are all equal.