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We are given two functions \( g(x) = -2x^2 + 4x + 1 \) and \( h(x) = \frac{1}{2}x^2 - x + \frac{11}{2} \). The problem asks for \( \lim_{x \to 1} f(x) \), where \( f(x) \) is a function that satisfies:

\[
g(x) \leq f(x) \leq h(x) \quad \text{for all } x.
\]

This situation hints at applying the **Squeeze Theorem** (also known as the Sandwich Theorem) because \( f(x) \) is "squeezed" between two functions, \( g(x) \) and \( h(x) \). To apply the Squeeze Theorem, we need to evaluate the limits of \( g(x) \) and \( h(x) \) as \( x \to 1 \) and see if they are equal.

### Step 1: Evaluate \( \lim_{x \to 1} g(x) \)
\[
g(x) = -2x^2 + 4x + 1
\]
Substituting \( x = 1 \):
\[
g(1) = -2(1)^2 + 4(1) + 1 = -2 + 4 + 1 = 3
\]
Thus, \( \lim_{x \to 1} g(x) = 3 \).

### Step 2: Evaluate \( \lim_{x \to 1} h(x) \)
\[
h(x) = \frac{1}{2}x^2 - x + \frac{11}{2}
\]
Substituting \( x = 1 \):
\[
h(1) = \frac{1}{2}(1)^2 - 1 + \frac{11}{2} = \frac{1}{2} - 1 + \frac{11}{2} = \frac{1}{2} + \frac{11}{2} - 1 = 6 - 1 = 5
\]
Thus, \( \lim_{x \to 1} h(x) = 3 \).

### Step 3: Apply the Squeeze Theorem
Since \( g(x) \leq f(x) \leq h(x) \) and both \( \lim_{x \to 1} g(x) = 3 \) and \( \lim_{x \to 1} h(x) = 3 \), by the Squeeze Theorem, we can conclude:
\[
\lim_{x \to 1} f(x) = 3
\]

Thus, the limit is \( \boxed{3} \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the general Squeeze Theorem, and when can it be applied?
2. How can we visually interpret the squeezing of a function between two bounds on a graph?
3. Could the limit change if \( g(x) \) and \( h(x) \) had different limits as \( x \to 1 \)?
4. How does the concept of continuity tie into the Squeeze Theorem?
5. What are the necessary conditions for the Squeeze Theorem to work?

**Tip:** The Squeeze Theorem is especially useful when you cannot directly compute the limit of a function but can bound it between two other functions whose limits are easier to find.

Let g and h be the functions defined by g(x) = -2x^2 + 4x + 1 and h(x) = (1/2)x^2 - x + 11/2. If f is a function that satisfies g(x) ≤ f(x) ≤ h(x) for all x, what is lim(x→1) f(x)?

Learn how to apply the Squeeze Theorem to evaluate the limit of a function bounded between two quadratic functions. This detailed solution demonstrates step-by-step evaluation for lim(x→1) f(x), where g(x) = -2x^2 + 4x + 1 and h(x) = (1/2)x^2 - x + 11/2.

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