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To solve the limit:

\[
\lim_{x \to 0} \frac{\sin(8x)}{\tan(2x)}
\]

We can use the small-angle approximation for sine and tangent, which states that as \(x \to 0\),

\[
\sin(x) \approx x \quad \text{and} \quad \tan(x) \approx x.
\]

Let's break it down step-by-step:

1. The given limit is:

\[
\lim_{x \to 0} \frac{\sin(8x)}{\tan(2x)}.
\]

2. Applying the small-angle approximations:

\[
\sin(8x) \approx 8x \quad \text{and} \quad \tan(2x) \approx 2x \quad \text{as} \quad x \to 0.
\]

3. Substituting these approximations into the original expression:

\[
\lim_{x \to 0} \frac{8x}{2x}.
\]

4. Simplifying the expression:

\[
\lim_{x \to 0} \frac{8x}{2x} = \lim_{x \to 0} 4 = 4.
\]

Thus, the value of the limit is:

\[
\boxed{4}.
\]

Would you like more details on any of these steps or have further questions?

Here are five related questions to expand on this topic:
1. What happens if you substitute other values of \(x\) in the limit equation?
2. How does L'Hopital's Rule apply to solving limits like this?
3. Can the small-angle approximations be used for larger values of \(x\)?
4. What is the geometric interpretation of the small-angle approximation for sine and tangent?
5. How does the limit change if you replace the functions with their hyperbolic counterparts, like \(\sinh(x)\) and \(\tanh(x)\)?

**Tip:** When solving limits involving trigonometric functions near zero, small-angle approximations are a powerful tool for simplifying expressions.

Learn how to solve the limit problem lim_{x → 0} sin(8x) / tan(2x) using small-angle approximations. This calculus problem involves trigonometric functions and simplifying expressions, suitable for high school students in grades 11-12.

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