The problem on your screen involves calculating a limit expression. It is given that:

$\lim_{x \to 3} f(x) = 4, \quad \lim_{x \to 3} g(x) = 3, \quad \lim_{x \to 3} h(x) = 2.$

You are asked to find:

$\lim_{x \to 3} \frac{8g(x) - 4h(x)}{(f(x))^2}.$

Step-by-step solution:

1. First, apply the limits of $g(x)$, $h(x)$, and $f(x)$ as $x$ approaches 3.

$\lim_{x \to 3} 8g(x) = 8 \times 3 = 24,$ $\lim_{x \to 3} 4h(x) = 4 \times 2 = 8,$ $\lim_{x \to 3} (f(x))^2 = (4)^2 = 16.$

2. Now substitute these limit values into the original expression:

$\lim_{x \to 3} \frac{8g(x) - 4h(x)}{(f(x))^2} = \frac{24 - 8}{16} = \frac{16}{16} = 1.$

Thus, the answer is 1.

Would you like further details or have any questions?

Related Questions:

1. How do you apply L'Hôpital's rule to more complex limits?
2. What is the meaning of the limit of a function as $x$ approaches a number?
3. How does the concept of continuity affect the calculation of limits?
4. What is the difference between one-sided and two-sided limits?
5. Can you explain the different types of indeterminate forms in limit problems?

Tip: When dealing with limits, always check if direct substitution works before trying other methods like L'Hôpital's rule.