Let’s evaluate the limits based on the graph provided.

Part (a)

$\lim_{x \to 2^-} f(x)$ This is the left-hand limit as $x$ approaches 2. From the graph, as $x$ approaches 2 from the left, the function $f(x)$ approaches 1.

So: $\lim_{x \to 2^-} f(x) = 1$

Part (b)

$\lim_{x \to 2^+} f(x)$ This is the right-hand limit as $x$ approaches 2. From the graph, as $x$ approaches 2 from the right, the function $f(x)$ approaches -1.

So: $\lim_{x \to 2^+} f(x) = -1$

Part (c)

$\lim_{x \to 2} f(x)$ The overall limit exists if and only if the left-hand limit equals the right-hand limit. Here, the left-hand limit is 1 and the right-hand limit is -1, so they are not equal. Therefore, the limit does not exist.

So: $\lim_{x \to 2} f(x) \text{ does not exist.}$

Summary of Results:

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

Let me know if you'd like a detailed explanation of why limits behave this way!

Related Questions:

1. What is the definition of left-hand and right-hand limits?
2. How do discontinuities affect the existence of limits?
3. How would you compute a limit if the graph is not provided?
4. How does the concept of limits relate to derivatives?
5. Can a function be continuous but have different left and right limits at a point?

Tip: When determining limits from graphs, always observe trends carefully as $x$ approaches from both sides.