https://file.aiquickdraw.com/mathbot/file/1727085527752eiaep5p6.jpg

Let's analyze the graph and the limits presented in the question.

The function \( f(x) \) is represented graphically, and the question asks us to evaluate the following limits:
1. \( \lim\limits_{x \to 2^-} f(x) \) (left-hand limit as \( x \) approaches 2)
2. \( \lim\limits_{x \to 2^+} f(x) \) (right-hand limit as \( x \) approaches 2)
3. \( \lim\limits_{x \to 2} f(x) \) (two-sided limit as \( x \) approaches 2)

### 1. \( \lim\limits_{x \to 2^-} f(x) \)
From the graph, as \( x \) approaches 2 from the left (i.e., \( x \to 2^- \)), the function value tends towards \( 3 \). Therefore:
\[
\lim_{x \to 2^-} f(x) = 3
\]

### 2. \( \lim\limits_{x \to 2^+} f(x) \)
As \( x \) approaches 2 from the right (i.e., \( x \to 2^+ \)), the function value tends towards \( 1 \). Therefore:
\[
\lim_{x \to 2^+} f(x) = 1
\]

### 3. \( \lim\limits_{x \to 2} f(x) \)
The two-sided limit exists only if the left-hand and right-hand limits are equal. In this case:
\[
\lim_{x \to 2^-} f(x) = 3 \quad \text{and} \quad \lim_{x \to 2^+} f(x) = 1
\]
Since these two limits are not equal, the two-sided limit does **not exist**.

### Final Answers:
1. \( \lim_{x \to 2^-} f(x) = 3 \)
2. \( \lim_{x \to 2^+} f(x) = 1 \)
3. \( \lim_{x \to 2} f(x) \) does not exist.

Would you like further details on any of the steps? 

Here are some related questions:
1. What is the importance of checking both left-hand and right-hand limits?
2. How can a function be continuous at a point if the two-sided limit does not exist?
3. What types of discontinuities does this graph show at \( x = 2 \)?
4. Can a function have different limits from the left and right but still be continuous?
5. How do removable and jump discontinuities differ?

**Tip**: Always check both the left-hand and right-hand limits when determining if a two-sided limit exists.

Let f(x) be described by the graph below. Evaluate the following limits, if they exist:
1. lim_{x→2^-} f(x)
2. lim_{x→2^+} f(x)
3. lim_{x→2} f(x)

Analyze the graph of f(x) and evaluate the left-hand limit, right-hand limit, and two-sided limit as x approaches 2. This problem covers key calculus concepts like limits, discontinuities, and understanding graphs of functions. Suitable for students in Grades 10-12.

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