https://file.aiquickdraw.com/mathbot/file/1726531265620hu2duuse.jpg

Let's analyze the graph and determine the one-sided and two-sided limits:

### 1. \(\lim\limits_{x \to 1^-} f(x)\)
This is the limit of \( f(x) \) as \( x \) approaches 1 from the left. As \( x \to 1^- \), the value of \( f(x) \) approaches -2.

\[
\lim\limits_{x \to 1^-} f(x) = -2
\]

### 2. \(\lim\limits_{x \to 1^+} f(x)\)
This is the limit of \( f(x) \) as \( x \) approaches 1 from the right. As \( x \to 1^+ \), the value of \( f(x) \) also approaches -2.

\[
\lim\limits_{x \to 1^+} f(x) = -2
\]

### 3. \(\lim\limits_{x \to 1} f(x)\)
Since both the left-hand limit and right-hand limit at \( x = 1 \) are equal, the two-sided limit exists and is equal to -2.

\[
\lim\limits_{x \to 1} f(x) = -2
\]

### 4. \(\lim\limits_{x \to 2} f(x)\)
At \( x = 2 \), the function has a hole, meaning the function does not exist at this point. However, the value of the function from both the left and right approaches -1.

\[
\lim\limits_{x \to 2} f(x) = -1
\]

### 5. \(\lim\limits_{x \to 3^-} f(x)\)
This is the limit as \( x \) approaches 3 from the left. As \( x \to 3^- \), the value of \( f(x) \) approaches 0.

\[
\lim\limits_{x \to 3^-} f(x) = 0
\]

### 6. \(\lim\limits_{x \to 3^+} f(x)\)
This is the limit as \( x \) approaches 3 from the right. As \( x \to 3^+ \), the value of \( f(x) \) approaches -1.

\[
\lim\limits_{x \to 3^+} f(x) = -1
\]

### 7. \(\lim\limits_{x \to 3} f(x)\)
Since the left-hand and right-hand limits as \( x \to 3 \) are different (0 from the left and -1 from the right), the two-sided limit does not exist (DNE).

\[
\lim\limits_{x \to 3} f(x) = \text{DNE}
\]

### 8. \(\lim\limits_{x \to 4} f(x)\)
As \( x \to 4 \), the value of \( f(x) \) approaches -1 from both sides.

\[
\lim\limits_{x \to 4} f(x) = -1
\]

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Would you like further clarification on any of these limits or additional explanations?

Here are five related questions:
1. What is the difference between one-sided and two-sided limits?
2. Can a limit exist at a point where the function is not defined?
3. How do we determine whether a two-sided limit exists?
4. What is the significance of discontinuities in limit calculations?
5. How do limits relate to the derivative of a function?

**Tip:** When analyzing limits graphically, always check both the left-hand and right-hand behavior at the point of interest.

The graph of f(x) given above. Use it to find the following one-sided and two-sided limits.

Learn how to analyze one-sided and two-sided limits from a graph. This problem focuses on determining the limits at various points, checking for continuity, and understanding where limits do not exist. Suitable for students in Grades 11-12 studying calculus.

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