Let's analyze the graph and solve the given questions step-by-step.

Given Problem

We are asked to:

1. Evaluate $f(1)$.
2. Evaluate $\lim_{x \to 3} f(x)$.
3. Evaluate $\lim_{x \to 4^+} f(x)$.
4. Evaluate $\lim_{x \to 8} f(x)$.

Step 1: $f(1)$

From the graph, at $x = 1$, the function $f(x)$ is at $y = 4$.

Thus, $f(1) = 4$.

Step 2: $\lim_{x \to 3} f(x)$

To find $\lim_{x \to 3} f(x)$, we examine the behavior of $f(x)$ as $x$ approaches 3 from both sides:

• As $x \to 3^-$: The function approaches $y = 2$ (from the left side).
• As $x \to 3^+$: The function approaches $y = 4$ (from the right side).

Since the left-hand limit ($2$) and right-hand limit ($4$) are not equal, the limit does not exist (DNE).

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

Step 3: $\lim_{x \to 4^+} f(x)$

To find $\lim_{x \to 4^+} f(x)$, we examine the behavior of $f(x)$ as $x$ approaches 4 from the right side:

• As $x \to 4^+$: The function immediately approaches $y = 3$.

Thus, $\lim_{x \to 4^+} f(x) = 3$.

Step 4: $\lim_{x \to 8} f(x)$

To find $\lim_{x \to 8} f(x)$, we examine the behavior of $f(x)$ as $x$ approaches 8 from both sides:

• As $x \to 8^-$: The function approaches $y = 2$ (from the left side).
• As $x \to 8^+$: The function approaches $y = 2$ (from the right side).

Since both sides agree, the limit exists and is $2$.

Thus, $\lim_{x \to 8} f(x) = 2$.

Final Answers:

1. $f(1) = 4$
2. $\lim_{x \to 3} f(x) = \text{DNE}$
3. $\lim_{x \to 4^+} f(x) = 3$
4. $\lim_{x \to 8} f(x) = 2$

Would you like a detailed explanation of any specific part? Let me know!

Related Questions:

1. How do we determine continuity of a function at a point from a graph?
2. What are the differences between left-hand and right-hand limits?
3. Why does a limit fail to exist at certain points?
4. How does the graph of a piecewise function relate to limits and continuity?
5. Can a function have a limit even if it is undefined at a point?

Tip:

When evaluating limits from a graph, always check the left-hand and right-hand behaviors carefully, especially at points of discontinuity!