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### Problem breakdown:
- Functions:
  \[
  f(x) = \frac{x}{4 - x^2} \quad \text{and} \quad g(x) = 4x
  \]
- The composite function is defined as:
  \[
  h(x) = f(g(x))
  \]
- We need to find \( \frac{h(4)}{g(1)} \).

### Step 1: Compute \( h(4) = f(g(4)) \)
We start by finding \( g(4) \):
\[
g(4) = 4 \times 4 = 16
\]
Now substitute \( g(4) \) into \( f(x) \):
\[
h(4) = f(16) = \frac{16}{4 - 16^2} = \frac{16}{4 - 256} = \frac{16}{-252} = -\frac{4}{63}
\]

### Step 2: Compute \( g(1) \)
\[
g(1) = 4 \times 1 = 4
\]

### Step 3: Find \( \frac{h(4)}{g(1)} \)
\[
\frac{h(4)}{g(1)} = \frac{-\frac{4}{63}}{4} = -\frac{4}{63} \times \frac{1}{4} = -\frac{1}{63}
\]

### Answer:
The correct answer is \( \boxed{a. -\frac{1}{63}} \).

Would you like more details or have any questions? Here are some related questions to explore:

1. How does the chain rule apply when differentiating composite functions like \( h(x) = f(g(x)) \)?
2. What is the general formula for calculating the derivative of \( f(g(x)) \)?
3. What are the conditions under which the function \( f(x) = \frac{x}{4 - x^2} \) is undefined?
4. How would the problem change if \( f(x) \) and \( g(x) \) were polynomials?
5. What are the applications of composite functions in real-world scenarios?

**Tip:** Always substitute and simplify before performing any calculations to avoid mistakes in multi-step problems.

Sejam f e g as funções reais de variável real definidas por f(x) = x / (4 - x^2) e g(x) = 4x. Se h é a função composta de f e g, isto é, h(x) = f(g(x)) então, o valor de h(4) / g(1) é?

Learn how to evaluate the composite function h(x) = f(g(x)) where f(x) = x / (4 - x^2) and g(x) = 4x. This problem focuses on algebraic manipulation and fraction evaluation. Suitable for advanced high school or early university students.

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