https://file.aiquickdraw.com/mathbot/file/1726827633837bfhnl6xf.jpg

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.

Misalkan fungsi f, g, dan h merupakan fungsi yang memenuhi sifat berikut: lim(x->3)f(x) = 4, lim(x->3)g(x) = 3, lim(x->3)h(x) = 2. Tentukanlah lim(x->3)[8g(x) - 4h(x)] / (f(x))^2

Learn how to calculate the limit of the expression [8g(x) - 4h(x)] / (f(x))^2 as x approaches 3, where lim(x->3)f(x) = 4, lim(x->3)g(x) = 3, and lim(x->3)h(x) = 2. This problem covers key algebraic and limit concepts. Suitable for Grades 10-12.

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