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.