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This problem involves applying the **Squeeze Theorem** to find the limit of \( f(x) \) as \( x \) approaches 48. Here's how we can solve it:

### Step 1: Analyze the given inequality

You are given that:
\[
\frac{Ax}{x+3} < f(x) < x + 768
\]
for all \( x \) near 48 but not equal to 48.

We need to use the Squeeze Theorem, which states that if a function is "squeezed" between two other functions that have the same limit at a point, then the function also has that limit.

### Step 2: Set up the limit as \( x \to 48 \)

Let’s calculate the limits of both the left and right sides of the inequality as \( x \to 48 \):

1. **Left-hand side:**
   \[
   \lim_{x \to 48} \frac{Ax}{x+3} = \frac{A \cdot 48}{48 + 3} = \frac{48A}{51}
   \]

2. **Right-hand side:**
   \[
   \lim_{x \to 48} (x + 768) = 48 + 768 = 816
   \]

Thus, for the Squeeze Theorem to apply, the left-hand side must approach 816 as well, meaning:
\[
\frac{48A}{51} = 816
\]

### Step 3: Solve for \( A \)

We now solve the equation:
\[
\frac{48A}{51} = 816
\]
Multiplying both sides by 51:
\[
48A = 816 \times 51
\]
\[
48A = 41616
\]
Dividing both sides by 48:
\[
A = \frac{41616}{48} = 867
\]

So, the value of \( A \) is **867**.

### Step 4: Apply the Squeeze Theorem to find the limit

Now that we know \( A = 867 \), we can conclude the limit of \( f(x) \) as \( x \to 48 \) by applying the Squeeze Theorem:
\[
\lim_{x \to 48} f(x) = 816
\]

### Summary of Answers:
(a) \( A = 867 \)  
(b) \( \lim_{x \to 48} f(x) = 816 \)

Let me know if you'd like further details or have any other questions!

### 5 Related Questions:
1. What are the conditions for the Squeeze Theorem to be applicable?
2. How can you use limits to determine the behavior of a function around a point?
3. What happens if the left-hand and right-hand limits in a squeeze inequality approach different values?
4. How does the value of \( A \) change the lower bound of the function in this inequality?
5. Can the Squeeze Theorem be used if the function is not continuous at the limit point?

### Tip:
Always ensure that the upper and lower bounds in the Squeeze Theorem converge to the same value before concluding that the function itself does as well.

Suppose that A is a constant and f(x) is a function of x such that (Ax / (x+3)) < f(x) < x + 768 for all x near 48 but not equal to 48. We are interested in finding the limit of f(x) as x approaches 48 by means of the Squeeze Theorem.

Learn how to apply the Squeeze Theorem to find the limit of f(x) as x approaches 48. The problem involves inequalities with f(x) bounded between (Ax / (x+3)) and x + 768, and solving for a constant A to make the Squeeze Theorem applicable.

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