https://file.aiquickdraw.com/mathbot/file/1726900202727l49sk80a.jpg

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.

The function f(x) is graphed above. Evaluate the following limits: 

a) lim(x→2⁺) f(x)

b) lim(x→−2) f(x)

c) lim(x→2) f(x)

d) lim(x→1) f(x)

Learn how to evaluate limits of a function f(x) using its graph at specific points like x→2⁺, x→-2, x→2, and x→1. This example covers concepts of limits, right-hand and left-hand limits, and continuity for students in Grades 11-12.

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