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We are tasked with using the Squeeze Theorem to evaluate \(\lim_{x \to 1} f(x)\), given that:

\[
-3x^4 + 12x^3 - 21x^2 + 20x - 9 \leq f(x) \leq 4x^4 - 16x^3 + 26x^2 - 18x + 3
\]

for all real numbers \(x\) near 1.

### Steps to Solve:

1. **Evaluate the lower bound limit:**
   \[
   \lim_{x \to 1} \left(-3x^4 + 12x^3 - 21x^2 + 20x - 9 \right)
   \]
   Substituting \(x = 1\):
   \[
   = -3(1)^4 + 12(1)^3 - 21(1)^2 + 20(1) - 9
   = -3 + 12 - 21 + 20 - 9
   = -1
   \]

2. **Evaluate the upper bound limit:**
   \[
   \lim_{x \to 1} \left( 4x^4 - 16x^3 + 26x^2 - 18x + 3 \right)
   \]
   Substituting \(x = 1\):
   \[
   = 4(1)^4 - 16(1)^3 + 26(1)^2 - 18(1) + 3
   = 4 - 16 + 26 - 18 + 3
   = -1
   \]

Since both the lower and upper bounds converge to \(-1\), the Squeeze Theorem tells us that:

\[
\lim_{x \to 1} f(x) = -1
\]

Would you like further clarification or details?

### Follow-up Questions:

1. What is the Squeeze Theorem and how is it applied?
2. Can the Squeeze Theorem be used for limits that approach infinity?
3. What happens if the upper and lower bounds do not converge to the same value?
4. How can this method be extended to more complex functions?
5. What are the conditions for using the Squeeze Theorem?

### Tip:
When using the Squeeze Theorem, always ensure that both the upper and lower functions have limits at the point of interest and that they squeeze the given function in a specific range.

Use the Squeeze Theorem to evaluate \( \lim_{x \to 1} f(x) \), given that \(-3x^4 + 12x^3 - 21x^2 + 20x - 9 \leq f(x) \leq 4x^4 - 16x^3 + 26x^2 - 18x + 3\) for all real numbers x near 1.

Learn how to use the Squeeze Theorem to evaluate \( \lim_{x \to 1} f(x) \) for the given inequality involving polynomials. This problem covers key calculus concepts, including limits and the Squeeze Theorem, suitable for students in advanced calculus.

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