https://file.aiquickdraw.com/mathbot/file/17264613533992o0vt7rn.jpg

The problem in the image presents the function \( f : \mathbb{R} \to \mathbb{R} \) and provides the following information:

\[
\lim_{x \to 1} f(x) = 2 \quad \text{and} \quad \lim_{x \to 1^+} f(x) = 3
\]
Then it asks which of the following statements must be true:

- **Statement A**: The limit of the function \( f \) as \( x \) approaches 1 exists.
- **Statement B**: The function \( f \) is not defined at \( x = 1 \).

### Analysis:

1. **Statement A**: For the limit of a function \( \lim_{x \to 1} f(x) \) to exist, the left-hand limit \( \lim_{x \to 1^-} f(x) \) and the right-hand limit \( \lim_{x \to 1^+} f(x) \) must both exist and be equal. Here, the right-hand limit \( \lim_{x \to 1^+} f(x) = 3 \), while \( \lim_{x \to 1^-} f(x) = 2 \). Since the left-hand and right-hand limits are not equal, the overall limit \( \lim_{x \to 1} f(x) \) does not exist.

   Therefore, **Statement A is false**.

2. **Statement B**: The problem does not provide direct information about whether the function is defined at \( x = 1 \), but since it specifies limits approaching \( x = 1 \) from both sides without defining \( f(1) \), we assume that \( f(1) \) may not be defined. This makes **Statement B plausible**.

Thus, the correct answer is **B only**.

### Would you like further clarification or have any questions?  
Here are 5 relative questions to expand on this:

1. What conditions must be met for a limit to exist at a point?
2. How do left-hand and right-hand limits relate to the existence of a two-sided limit?
3. Can a function have different limits from the left and right at the same point?
4. What does it mean for a function to be undefined at a point?
5. Can a function be continuous if the limit exists but the function is not defined at that point?

**Tip**: When evaluating limits, always check whether the left-hand and right-hand limits match before concluding that the limit exists.

Consider any function f : ℝ → ℝ, where lim f(x) = 2 and lim f(x) = 3 as x approaches 1. Which of the following statements about f must be true? Statement A: The limit of the function f as x approaches 1 exists. Statement B: The function f is not defined at x = 1.

This problem explores limits as x approaches 1 for a function f(x). Students will determine whether the limit exists and assess if the function is defined at x = 1. Key concepts include left-hand and right-hand limits, conditions for the existence of limits, and piecewise functions.

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