Let's analyze the graph of $y = f(x)$ to answer the given questions.

1. $f(6) = 4$ (already provided).

Now, let's address the limit questions:

2. $\lim_{x \to 4} f(x)$:

   The graph shows a discontinuity at $x = 4$. From the left side of $x = 4$ (as $x \to 4^-$), the value of $f(x)$ approaches 1. From the right side (as $x \to 4^+$), the value of $f(x)$ approaches 3. Since the left-hand limit and the right-hand limit are not equal, the overall limit does not exist.

   Therefore, $\lim_{x \to 4} f(x) = \text{DNE}$.

3. $\lim_{x \to 1^+} f(x)$:

   Looking at the graph from the right of $x = 1$, the value of $f(x)$ is 5. Since the graph approaches $x = 1$ from the right with no discontinuity, the right-hand limit exists and equals 5.

   Therefore, $\lim_{x \to 1^+} f(x) = 5$.

4. $\lim_{x \to 3} f(x)$:

   At $x = 3$, there is also a discontinuity. From the left of $x = 3$, the value of $f(x)$ approaches 2, and from the right, the value of $f(x)$ approaches 4. Since these values are different, the limit does not exist.

   Therefore, $\lim_{x \to 3} f(x) = \text{DNE}$.

Summary of Answers:

• $f(6) = 4$
• $\lim_{x \to 4} f(x) = \text{DNE}$
• $\lim_{x \to 1^+} f(x) = 5$
• $\lim_{x \to 3} f(x) = \text{DNE}$

Would you like further details or explanations on any of these steps?

Here are 5 additional questions that expand on this:

1. What is $\lim_{x \to 5} f(x)$?
2. What is the value of $f(0)$?
3. Is $f(x)$ continuous at $x = 2$?
4. What is the left-hand limit $\lim_{x \to 1^-} f(x)$?
5. How many discontinuities does the function $f(x)$ have on the interval $[0,9]$?

Tip: Limits involving discontinuities are best handled by checking the behavior from both sides (left and right) to confirm if they match.