Let's go through the graph and analyze each part of the question to determine the answers.

Given:

The graph depicts the function $f(x)$. We need to determine limits and function values for various conditions.

(a) $\lim_{x \to 2^-} f(x)$

This asks for the left-hand limit as $x$ approaches 2. From the graph, as $x$ approaches 2 from the left, $f(x)$ approaches 1.

So, the answer for part (a) is: $\lim_{x \to 2^-} f(x) = 1$

(b) $\lim_{x \to 2^+} f(x)$

This is the right-hand limit as $x$ approaches 2. From the graph, as $x$ approaches 2 from the right, $f(x)$ approaches 3.

So, the answer for part (b) is: $\lim_{x \to 2^+} f(x) = 3$

(c) $\lim_{x \to 2} f(x)$

Since the left-hand and right-hand limits as $x \to 2$ are not equal (1 from the left and 3 from the right), the overall limit does not exist.

So, the answer for part (c) is: $\lim_{x \to 2} f(x) = \text{DNE}$

(d) $f(2)$

From the graph, the function value at $x = 2$ is not defined, as there is no closed dot at $x = 2$.

So, the answer for part (d) is: $f(2) = \text{DNE}$

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

As $x$ approaches 4, the function value smoothly approaches $f(x) = 2$. Both the left-hand and right-hand limits at $x = 4$ exist and are equal to 2.

So, the answer for part (e) is: $\lim_{x \to 4} f(x) = 2$

(f) $f(4)$

From the graph, at $x = 4$, there is a closed dot at $f(4) = 2$.

So, the answer for part (f) is: $f(4) = 2$

Summary of Answers:

• (a) $\lim_{x \to 2^-} f(x) = 1$
• (b) $\lim_{x \to 2^+} f(x) = 3$
• (c) $\lim_{x \to 2} f(x) = \text{DNE}$
• (d) $f(2) = \text{DNE}$
• (e) $\lim_{x \to 4} f(x) = 2$
• (f) $f(4) = 2$

Would you like further details or clarifications on any part of this solution? Here are some additional questions that might interest you:

1. How do left-hand and right-hand limits affect overall limits?
2. Can the function have a limit at a point where it is undefined?
3. What does it mean if a limit does not exist?
4. How can we tell from a graph if a function is continuous at a point?
5. What are the differences between limits and actual function values?

Tip: When both left-hand and right-hand limits are equal at a point, the overall limit exists at that point. Otherwise, the limit does not exist.