Let's solve the given limits step by step using the provided graph.

(a) $\lim_{x \to -2^-} f(x)$

• This represents the left-hand limit as $x$ approaches $-2$.
• From the graph, observe the behavior of $f(x)$ as $x$ approaches $-2$ from the left. The graph approaches $y = 1$.
• Answer: $\lim_{x \to -2^-} f(x) = 1$

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

• This represents the right-hand limit as $x$ approaches $-2$.
• From the graph, observe the behavior of $f(x)$ as $x$ approaches $-2$ from the right. The graph also approaches $y = 1$.
• Answer: $\lim_{x \to -2^+} f(x) = 1$

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

• Here, $x$ approaches $3$ from both sides (two-sided limit).
• From the graph, as $x$ approaches $3$, the function $f(x)$ does not approach a single value because there is a discontinuity:
  • The graph shows an open circle at $y = -1$ (indicating the function does not take that value).
  • There is a filled circle at $y = 2$, which is the actual value of $f(3)$.
• Since the left-hand limit and right-hand limit do not match, the two-sided limit does not exist.
• Answer: $\lim_{x \to 3} f(x) \text{ does not exist.}$

Final Answers:

1. (a) $\lim_{x \to -2^-} f(x) = 1$
2. (b) $\lim_{x \to -2^+} f(x) = 1$
3. (c) $\lim_{x \to 3} f(x) = \text{Does not exist}$

Would you like me to clarify further or explore related examples?

5 Related Questions:

1. What are left-hand and right-hand limits? How are they defined?
2. How can you determine if a function is continuous at a point using limits?
3. How do open and closed circles on a graph affect limit evaluation?
4. What is the difference between limits and actual function values at a point?
5. How can we identify when a limit does not exist?

Tip: When evaluating limits graphically, carefully observe the behavior of the function as it approaches the point from both sides. If they match, the limit exists!