suppose -2 <= f(x) <= x^2-4x+2 for all x = cannot equal 2
Which of the following MUST be true? Select all correct answer choices.

1. lim x-> 2 f(x) = -2
2. f(2) = -2
3. lim x -> 0^- f(x) = lim x -> 0^+ f(x)
4. lim x-> 0 f(x) does not exist 

We are given that \( -2 \leq f(x) \leq x^2 - 4x + 2 \) for all \( x \neq 2 \), and we need to determine which of the statements must be true. Let's analyze each option.

### Option 1: \( \lim_{x \to 2} f(x) = -2 \)

- We know that \( f(x) \) is bounded between \(-2\) and \(x^2 - 4x + 2\), but no additional information about the behavior of \( f(x) \) as \( x \) approaches 2 is given.
- Without more details about the behavior of \( f(x) \) near \( x = 2 \), this limit does not have to equal \(-2\). The upper bound \( x^2 - 4x + 2 \) evaluated at \( x = 2 \) is also \(-2\), but this alone is not sufficient to conclude the limit equals \(-2\).
  
Thus, **Option 1 is not necessarily true.**

### Option 2: \( f(2) = -2 \)

- The condition \( f(x) \) is defined for all \( x \neq 2 \), and we are not given the value of \( f(x) \) at \( x = 2 \). Therefore, there is no information to confirm that \( f(2) = -2 \).
  
Thus, **Option 2 is not necessarily true** because we don't know if \( f(x) \) is even defined at \( x = 2 \).

### Option 3: \( \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) \)

- The function \( f(x) \) is continuous for all \( x \neq 2 \), and there is no discontinuity mentioned at \( x = 0 \).
- Since \( f(x) \) is bounded by a continuous function \( x^2 - 4x + 2 \), the left-hand limit (\( \lim_{x \to 0^-} f(x) \)) and right-hand limit (\( \lim_{x \to 0^+} f(x) \)) must agree at \( x = 0 \), as there is no indication of a discontinuity at this point.

Thus, **Option 3 is true**.

### Option 4: \( \lim_{x \to 0} f(x) \) does not exist

- As discussed in Option 3, since \( f(x) \) is bounded by a continuous function and there is no indication of a discontinuity at \( x = 0 \), the limit \( \lim_{x \to 0} f(x) \) must exist.
  
Thus, **Option 4 is false**.

### Conclusion:

The correct answer is:
- **Option 3: \( \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) \)**

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suppose -2 <= f(x) <= x^2-4x+2 for all x ≠ 2. Which of the following MUST be true? Select all correct answer choices.
1. lim x-> 2 f(x) = -2
2. f(2) = -2
3. lim x -> 0^- f(x) = lim x -> 0^+ f(x)
4. lim x-> 0 f(x) does not exist

Explore a problem involving limits, inequalities, and continuity for a function f(x) bounded by -2 <= f(x) <= x^2 - 4x + 2. Analyze key limit behavior and discontinuities, using the Squeeze Theorem. Suitable for students in Grades 11-12.

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