https://file.aiquickdraw.com/mathbot/file/17263805918686v93zxfj.jpg

Let's go through the question from the image and solve it step by step. 

### The problem:
We are given the graph of a function \( f(x) \), and we need to determine specific limit values from the graph:

#### (a) \(\lim_{x \to 1} f(x)\)
This asks for the limit of the function as \(x\) approaches 1.

#### (b) \(\lim_{x \to 3^-} f(x)\)
This is the left-hand limit as \(x\) approaches 3.

#### (c) \(\lim_{x \to 3^+} f(x)\)
This is the right-hand limit as \(x\) approaches 3.

#### (d) \(\lim_{x \to 3} f(x)\)
This is the overall limit as \(x\) approaches 3 (exists if the left-hand and right-hand limits are equal).

#### (e) \(f(3)\)
This is the value of the function at \(x = 3\).

### Analysis of the graph:
1. **(a) \(\lim_{x \to 1} f(x)\)**:
   The graph appears continuous around \(x = 1\). As \(x\) approaches 1, the value of \(f(x)\) tends to 2. Therefore, 
   \[
   \lim_{x \to 1} f(x) = 2.
   \]

2. **(b) \(\lim_{x \to 3^-} f(x)\)**:
   The left-hand limit as \(x\) approaches 3 is the value that the function approaches as \(x\) gets close to 3 from the left. From the graph, we see that as \(x\) approaches 3 from the left, the function value approaches 2. Thus,
   \[
   \lim_{x \to 3^-} f(x) = 2.
   \]

3. **(c) \(\lim_{x \to 3^+} f(x)\)**:
   The right-hand limit as \(x\) approaches 3 is the value that the function approaches as \(x\) gets close to 3 from the right. From the graph, the function value as \(x\) approaches 3 from the right is 4. Thus,
   \[
   \lim_{x \to 3^+} f(x) = 4.
   \]

4. **(d) \(\lim_{x \to 3} f(x)\)**:
   The overall limit exists if the left-hand limit and the right-hand limit are the same. Here, the left-hand limit is 2, and the right-hand limit is 4. Since they are not equal, the overall limit does not exist.
   \[
   \lim_{x \to 3} f(x) \text{ does not exist}.
   \]

5. **(e) \(f(3)\)**:
   To find \(f(3)\), we look at the value of the function exactly at \(x = 3\). From the graph, there is a filled dot at \(f(3) = 2\), indicating that the function's value at \(x = 3\) is 2. Thus,
   \[
   f(3) = 2.
   \]

### Final answers:
- (a) \(\lim_{x \to 1} f(x) = 2\)
- (b) \(\lim_{x \to 3^-} f(x) = 2\)
- (c) \(\lim_{x \to 3^+} f(x) = 4\)
- (d) \(\lim_{x \to 3} f(x)\) does not exist
- (e) \(f(3) = 2\)

Would you like a deeper explanation of any of these steps?

Here are 5 related questions you could explore:
1. What is the difference between left-hand and right-hand limits?
2. How can discontinuities in a function affect the existence of limits?
3. Can a function be continuous at a point where the limit does not exist?
4. How would a removable discontinuity affect the limit of a function?
5. What is the graphical interpretation of a function's value versus its limit at a point?

**Tip:** A limit exists at a point if and only if the left-hand and right-hand limits are equal at that point.

For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.

This lesson explains how to determine limits from a graph. It includes left-hand and right-hand limits and the function value at specific points. The problem is ideal for students studying advanced pre-calculus or calculus.

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