The problem provided asks to use the Squeeze Theorem to determine the limit of a function $f(x)$ as $x$ approaches 2.

We are given:

• $g(x) = -x^2 + x$
• $h(x) = x^2 - x$
• $g(x) \leq f(x) \leq h(x)$ for $1 \leq x \leq 3$

The Squeeze Theorem states that if $g(x) \leq f(x) \leq h(x)$ for all $x$ in some interval around $c$, except possibly at $c$ itself, and if

$\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$

then

$\lim_{x \to c} f(x) = L.$

Now, let's calculate the limits of $g(x)$ and $h(x)$ as $x$ approaches 2.

1. Limit of $g(x)$ as $x \to 2$:

$g(x) = -x^2 + x$

$\lim_{x \to 2} g(x) = -(2)^2 + 2 = -4 + 2 = -2$

2. Limit of $h(x)$ as $x \to 2$:

$h(x) = x^2 - x$

$\lim_{x \to 2} h(x) = (2)^2 - 2 = 4 - 2 = 2$

Since the limits of $g(x)$ and $h(x)$ as $x$ approaches 2 are not equal (i.e., -2 and 2 respectively), the Squeeze Theorem cannot be applied directly. Therefore, the limit of $f(x)$ as $x$ approaches 2 cannot be determined using the Squeeze Theorem.

The correct answer is "Cannot be determined".

Would you like any further details or have any questions?

Here are five related questions for further practice:

1. What conditions must be satisfied for the Squeeze Theorem to be applied successfully?
2. How would you determine the limit of a function that doesn't satisfy the conditions of the Squeeze Theorem?
3. What are some examples of functions where the Squeeze Theorem is typically used?
4. How can you use the definition of a limit to verify whether the Squeeze Theorem can be applied?
5. What are the implications if the limits of $g(x)$ and $h(x)$ are not equal at a point?

Tip: When using the Squeeze Theorem, ensure that both bounding functions converge to the same limit as $x$ approaches the point of interest.