Let's analyze the graph and answer the questions step by step based on the limits and function values.

(a) $\lim_{x \to 1} f(x)$

• From the graph, as $x$ approaches 1 from both sides, the function value approaches 2. Thus: $\lim_{x \to 1} f(x) = 2$

(b) $\lim_{x \to 3^-} f(x)$

• As $x$ approaches 3 from the left, the function value approaches 1. Thus: $\lim_{x \to 3^-} f(x) = 1$

(c) $\lim_{x \to 3^+} f(x)$

• From the graph, as $x$ approaches 3 from the right, the function seems to approach a value different from 1. It appears to approach 3. Thus: $\lim_{x \to 3^+} f(x) = 3$ So, the correct answer for part (c) should be 3, not what was originally selected.

(d) $\lim_{x \to 3} f(x)$

• Since the left-hand limit $\lim_{x \to 3^-} f(x) = 1$ and the right-hand limit $\lim_{x \to 3^+} f(x) = 3$, the limit does not exist because the left and right limits are not equal. Therefore: $\lim_{x \to 3} f(x) = \text{DNE (Does Not Exist)}$

(e) $f(3)$

• From the graph, at $x = 3$, there is a filled-in point at $y = 4$, indicating that $f(3) = 4$. Therefore: $f(3) = 4$

The only corrections needed are for parts (c) and (e):

• $\lim_{x \to 3^+} f(x) = 3$
• $f(3) = 4$

Do you want further clarification or details? Here are some related questions to deepen your understanding:

1. What is the difference between left-hand and right-hand limits?
2. Why does the limit not exist at $x = 3$?
3. How do we determine the exact function value $f(x)$ at a given point?
4. What are removable discontinuities, and does this graph contain any?
5. How can you verify limits algebraically if you are given a function formula instead of a graph?

Tip: When dealing with limits from graphs, always check whether the left-hand limit equals the right-hand limit to determine if the overall limit exists at a point.