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.