Let's analyze the mathematical content from the image. We have the following inequality for $f(x)$, a function of $x$:

$\frac{Ax}{x - 2} < f(x) < x - 512$

The goal is to find the constant $A$ so that the Squeeze Theorem can be applied to find the limit of $f(x)$ as $x$ approaches 32.

Step-by-step process:

1. Apply the Squeeze Theorem: According to the Squeeze Theorem, if we have a function $g(x) \leq f(x) \leq h(x)$ and both $g(x)$ and $h(x)$ approach the same limit $L$ as $x$ approaches some value, then $\lim_{x \to c} f(x) = L$.

2. Lower and Upper Bound Functions: The lower bound is $\frac{Ax}{x - 2}$, and the upper bound is $x - 512$. We need to compute the limits of both bounding functions as $x \to 32$.

3. Limit of the Upper Bound: $\lim_{x \to 32} (x - 512) = 32 - 512 = -480$

4. Limit of the Lower Bound: $\lim_{x \to 32} \frac{Ax}{x - 2}$ Plugging $x = 32$ into the expression: $\frac{Ax}{x - 2} = \frac{A \cdot 32}{32 - 2} = \frac{32A}{30}$ For the Squeeze Theorem to hold, this limit must also equal $-480$ (the same as the upper bound).

5. Solve for $A$: Set the lower bound equal to $-480$ and solve for $A$: $\frac{32A}{30} = -480$ Multiply both sides by 30: $32A = -480 \times 30 = -14400$ Divide by 32: $A = \frac{-14400}{32} = -450$

Thus, the correct value of $A$ is $-450$, not 30 as indicated in the image.

Would you like more details or explanations about this solution?

Here are some related questions for further practice:

1. How does the Squeeze Theorem work when one of the bounds tends to infinity?
2. What happens if the limits of the bounding functions don't match?
3. Can the Squeeze Theorem be applied if one of the bounding functions has a discontinuity?
4. How does changing the value of $A$ affect the behavior of $f(x)$?
5. What other methods can be used to find the limit of $f(x)$ as $x \to 32$ besides the Squeeze Theorem?

Tip: When applying the Squeeze Theorem, always verify that both bounds approach the same limit before concluding the behavior of the function.