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We are tasked with finding \(\lim_{x \to 1} g(x)\), given the inequality:

\[
2x \leq g(x) \leq x^4 - x^2 + 2 \quad \text{for all} \quad x.
\]

### Step 1: Analyzing the limit using the Squeeze Theorem
The Squeeze Theorem can be applied here because \(g(x)\) is squeezed between two functions, \(2x\) and \(x^4 - x^2 + 2\).

We will evaluate the limits of these bounding functions as \(x \to 1\):

#### Lower Bound: \(\lim_{x \to 1} 2x\)
\[
\lim_{x \to 1} 2x = 2(1) = 2.
\]

#### Upper Bound: \(\lim_{x \to 1} x^4 - x^2 + 2\)
\[
\lim_{x \to 1} (x^4 - x^2 + 2) = (1^4 - 1^2 + 2) = 1 - 1 + 2 = 2.
\]

### Step 2: Applying the Squeeze Theorem
Since both the lower bound and upper bound approach 2 as \(x \to 1\), by the Squeeze Theorem, the limit of \(g(x)\) must also approach 2:

\[
\lim_{x \to 1} g(x) = 2.
\]

### Final Answer:
\[
\lim_{x \to 1} g(x) = 2.
\]

Would you like further details or clarifications?

Here are 5 related questions to deepen your understanding:
1. How does the Squeeze Theorem work in other cases?
2. What are the conditions for applying the Squeeze Theorem?
3. Can \(g(x)\) be uniquely determined from the given inequality?
4. What is the significance of limit evaluation in calculus?
5. How would the solution change if the bounds for \(g(x)\) were different?

**Tip:** Always check that the limits of the bounding functions converge to the same value when using the Squeeze Theorem.

If 2x ≤ g(x) ≤ x^4 - x^2 + 2 for all x, evaluate lim(x→1) g(x).

Find the limit of g(x) as x approaches 1, given the inequality 2x ≤ g(x) ≤ x^4 - x^2 + 2. Learn how to apply the Squeeze Theorem, a crucial concept in calculus, to solve this problem. This solution is ideal for students in Grades 11-12.

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