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

a) $\lim_{x \to 2^+} f(x) = 0$

This is the right-hand limit as $x$ approaches 2. On the graph, as $x$ approaches 2 from the right, the value of $f(x)$ appears to approach 0. Therefore, the right-hand limit is indeed:

$\lim_{x \to 2^+} f(x) = 0$

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

Here, we are asked for the two-sided limit as $x$ approaches -2. From the graph:

• As $x$ approaches -2 from the left, $f(x)$ approaches -4.
• As $x$ approaches -2 from the right, $f(x)$ also approaches -4.

Since both the left-hand and right-hand limits agree, the limit exists, and we have:

$\lim_{x \to -2} f(x) = -4$

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

This asks for the two-sided limit as $x$ approaches 2.

• As $x$ approaches 2 from the left, $f(x)$ approaches 4.
• As $x$ approaches 2 from the right, $f(x)$ approaches 0.

Since the left-hand and right-hand limits do not match, the two-sided limit does not exist. Therefore:

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

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

Looking at the graph as $x$ approaches 1, both from the left and right, $f(x)$ is approaching -1. Hence:

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

Summary of answers:

a) $\lim_{x \to 2^+} f(x) = 0$
b) $\lim_{x \to -2} f(x) = -4$
c) $\lim_{x \to 2} f(x) \text{ does not exist}$
d) $\lim_{x \to 1} f(x) = -1$

Let me know if you need further details or clarification! Here are five related questions to expand on this:

1. What is the left-hand limit as $x \to 2^-$ for $f(x)$?
2. How do left-hand and right-hand limits relate to continuity at a point?
3. What is the definition of a two-sided limit and when does it exist?
4. Can a function have different left-hand and right-hand limits but still be defined at that point?
5. How do limits relate to the derivative of a function?

Tip: When evaluating a two-sided limit, check both the left-hand and right-hand limits separately. If they are not equal, the two-sided limit does not exist.