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The given problem asks to evaluate which of the following statements about a function \( f(x) \) are true, under the condition:

\[
-2 \leq f(x) \leq x^2 - 6x + 2 \quad \text{for all} \quad x \geq 2.
\]

The choices are:

1. \( \lim_{x \to \infty} f(x) = -2 \)
2. \( f(2) = 2 \)
3. \( \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{f(x)}{x} \)
4. \( \lim_{x \to \infty} f(x) \) does not exist.

### Analysis:

1. **Bounding \( f(x) \):** 
   - The function is bounded between \( -2 \) and \( x^2 - 6x + 2 \), so as \( x \to \infty \), we need to analyze the bounds' behavior:
     \[
     \lim_{x \to \infty} x^2 - 6x + 2 = \infty,
     \]
     which means \( f(x) \to \infty \) or could stay bounded within the range, depending on the nature of \( f(x) \).

2. **First option

Suppose −2 ≤ f(x) ≤ x² − 6x + 2 for all x ≥ 2. Which of the following MUST be true? Select all correct answer choices.

Explore the problem of finding the behavior of the function f(x) bounded by −2 ≤ f(x) ≤ x² − 6x + 2 for x ≥ 2. This analysis uses limits and the Squeeze Theorem to determine which statements about the function must be true.

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