https://file.aiquickdraw.com/mathbot/file/1727453770461t4wkssfz.jpg

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.

Use the given graph of f to state the value of each quantity, if it exists. (If an answer does not exist, enter DNE).

Explore how to determine the values of limits and function values from a given graph of f(x). Learn about left-hand and right-hand limits, continuity, and how to assess function values at specific points. Suitable for Grades 11-12 students.

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