https://file.aiquickdraw.com/mathbot/file/1727198258708wm48nnhn.jpg

Let's analyze the graph and solve each part step by step. 

We are looking at the limits as \( x \) approaches 3 from both directions and also at the actual value of \( f(3) \). The open and closed circles in the graph will help us determine these values.

### Part a: \(\lim_{x \to 3^-} f(x)\)
This represents the limit of \( f(x) \) as \( x \) approaches 3 from the left (negative direction). As we follow the graph from the left towards \( x = 3 \), we see that the function approaches \( y = 2 \). 

Thus, 
\[
\lim_{x \to 3^-} f(x) = 2
\]

### Part b: \(\lim_{x \to 3^+} f(x)\)
This is the limit of \( f(x) \) as \( x \) approaches 3 from the right (positive direction). As we follow the graph from the right towards \( x = 3 \), we observe that the function approaches \( y = 4 \).

Therefore,
\[
\lim_{x \to 3^+} f(x) = 4
\]

### Part c: \(\lim_{x \to 3} f(x)\)
The overall limit as \( x \) approaches 3 exists only if the left-hand limit and right-hand limit are equal. Since \( \lim_{x \to 3^-} f(x) = 2 \) and \( \lim_{x \to 3^+} f(x) = 4 \), the left-hand and right-hand limits are not equal. Thus, the limit does not exist.

So, 
\[
\lim_{x \to 3} f(x) \text{ does not exist}.
\]

### Part d: \( f(3) \)
The value of \( f(x) \) at \( x = 3 \) is indicated by the filled or open circle at \( x = 3 \). There is a closed (filled) circle at \( (3, 0) \), meaning that \( f(3) = 0 \).

Thus, 
\[
f(3) = 0.
\]

### Final answers:
- a) \(\lim_{x \to 3^-} f(x) = 2\)
- b) \(\lim_{x \to 3^+} f(x) = 4\)
- c) \(\lim_{x \to 3} f(x)\) does not exist
- d) \(f(3) = 0\)

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Would you like further explanation on any part of this solution?

Here are 5 related questions to extend your understanding:
1. How do you define the left-hand and right-hand limits in calculus?
2. What conditions must be met for a limit to exist at a point?
3. How do you interpret open and closed circles in graphs of functions?
4. Can a function be continuous at a point if the left-hand and right-hand limits are not equal?
5. What is the significance of a function's value at a point if the limit doesn't exist?

**Tip:** A limit may exist at a point even if the function is not defined at that point. Keep an eye on both the limit and the actual function value when analyzing graphs.

Use the graph below to determine the limits below.

Learn how to determine limits as x approaches 3 from both sides using a graph. This problem focuses on left-hand and right-hand limits, as well as evaluating the function value at x = 3, with a step-by-step explanation. Suitable for high school students in Grades 10-12.

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