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.