Let's evaluate each of the expressions using the graph provided.

a. $f(-2)$

• From the graph, at $x = -2$, the closed dot indicates $f(-2) = 0$.

b. $\lim_{x \to -2^+} f(x)$

• The right-hand limit as $x$ approaches $-2$ from the right is approaching $2$, based on the graph.

c. $\lim_{x \to -2} f(x)$

• For the two-sided limit, $\lim_{x \to -2} f(x)$, we must check both the left-hand and right-hand limits.
  • Left-hand limit: as $x$ approaches $-2$ from the left, the value approaches 0.
  • Right-hand limit: as $x$ approaches $-2$ from the right, the value approaches 2.
• Since the left-hand and right-hand limits do not agree, the two-sided limit does not exist.

d. $\lim_{x \to -1^+} f(x)$

• From the right, as $x \to -1$, the value of $f(x)$ approaches 0. So, $\lim_{x \to -1^+} f(x) = 0$.

e. $\lim_{x \to -1^-} f(x)$

• From the left, as $x \to -1$, the value of $f(x)$ approaches 0 as well. Hence, $\lim_{x \to -1^-} f(x) = 0$.

f. $\lim_{x \to -1} f(x)$

• Since both the left-hand and right-hand limits are equal to 0, $\lim_{x \to -1} f(x) = 0$.

g. $\lim_{x \to 1^+} f(x)$

• As $x$ approaches 1 from the right, the value of $f(x)$ approaches 0. Hence, $\lim_{x \to 1^+} f(x) = 0$.

h. $\lim_{x \to 1^-} f(x)$

• As $x$ approaches 1 from the left, the value of $f(x)$ approaches 0. Hence, $\lim_{x \to 1^-} f(x) = 0$.

i. $\lim_{x \to 1} f(x)$

• Since both left-hand and right-hand limits are equal, $\lim_{x \to 1} f(x) = 0$.

j. $f(3)$

• At $x = 3$, there is a closed dot at $f(3) = 5$. So, $f(3) = 5$.

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

• As $x$ approaches 3 from the right, the graph approaches $0$. Hence, $\lim_{x \to 3^+} f(x) = 0$.

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

• As $x$ approaches 3 from the left, the graph approaches $5$. Hence, $\lim_{x \to 3^-} f(x) = 5$.

m. $\lim_{x \to 3} f(x)$

• Since the left-hand limit ($5$) and right-hand limit ($0$) do not agree, $\lim_{x \to 3} f(x)$ does not exist.

Would you like more details or clarification on any of these points?

Here are some related questions to expand understanding:

1. How do you determine the existence of a limit at a point?
2. What is the significance of the right-hand and left-hand limits in relation to two-sided limits?
3. What role do discontinuities play in the behavior of limits?
4. Can a function have different values for left-hand and right-hand limits but still be continuous?
5. How would the graph change if the limit at $x = 3$ existed?

Tip: When evaluating limits from a graph, always check the behavior of the function from both sides of the point to see if they match.