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We are tasked with finding the limit:

\[
\lim_{z \to 0} h(z), \quad \text{where} \quad h(z) = \frac{\tan^3(4z)}{4z^2 \sin(z)}
\]

To solve this, we will use Taylor series expansions around \( z = 0 \) for both \( \tan(4z) \) and \( \sin(z) \), and then analyze the behavior as \( z \to 0 \).

### Step 1: Taylor Series Expansion
- **Expansion of \( \tan(4z) \):**

Using the Taylor series for \( \tan(x) \) around \( x = 0 \), we have:

\[
\tan(4z) = 4z + \frac{(4z)^3}{3} + O(z^5)
\]
Thus, \( \tan^3(4z) \) becomes:
\[
\tan^3(4z) = \left(4z + \frac{(4z)^3}{3} + O(z^5)\right)^3 = 64z^3 + O(z^5)
\]

- **Expansion of \( \sin(z) \):**

Using the Taylor series for \( \sin(z) \), we have:
\[
\sin(z) = z - \frac{z^3}{6} + O(z^5)
\]

### Step 2: Simplifying the limit expression
Now substitute these expansions into the original function \( h(z) \):

\[
h(z) = \frac{64z^3 + O(z^5)}{4z^2 (z - \frac{z^3}{6} + O(z^5))}
\]

First, simplify the denominator:

\[
4z^2 \sin(z) = 4z^3 - \frac{4z^5}{6} + O(z^7)
\]

Thus, the expression for \( h(z) \) becomes:

\[
h(z) = \frac{64z^3 + O(z^5)}{4z^3 - \frac{4z^5}{6} + O(z^7)}
\]

### Step 3: Evaluating the limit
As \( z \to 0 \), the leading terms in both the numerator and the denominator are \( 64z^3 \) and \( 4z^3 \), respectively. So, the limit simplifies to:

\[
\lim_{z \to 0} h(z) = \lim_{z \to 0} \frac{64z^3}{4z^3} = \frac{64}{4} = 16
\]

Thus, the desired limit is:

\[
\boxed{16}
\]

Let me know if you would like further details or clarifications!

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Here are 5 related questions to deepen your understanding:
1. How do Taylor series expansions help in solving limits involving trigonometric functions?
2. What other methods, besides Taylor expansion, could be used to solve this limit?
3. How does the limit of \( \tan(4z)/z \) behave as \( z \to 0 \)?
4. Can we apply L'Hopital's Rule to solve this limit? If so, how many derivatives would be needed?
5. What is the Taylor series expansion of \( \cos(z) \) around \( z = 0 \)?

**Tip:** Taylor expansions are powerful for handling indeterminate forms like \( 0/0 \). When faced with such forms, expansions around critical points (like 0) often lead to simplifications.

Let h(z) = (tan^3(4z)) / (4z^2 sin(z)). Find lim h(z) as z approaches 0.

Explore how to evaluate the limit of h(z) = (tan^3(4z)) / (4z^2 sin(z)) as z approaches 0 using Taylor series expansions. This step-by-step solution highlights key calculus concepts, including trigonometric expansions and simplifications. Suitable for advanced high school math students.

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