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Kerjakan soal berikut menggunakan rumus limit trigonometri 

The problem presented is a limit involving a trigonometric function:

\[
\lim_{{x \to 0}} \frac{\sin(2x)}{3x}
\]

To solve this, we will use the standard limit identity for small angles:

\[
\lim_{{x \to 0}} \frac{\sin(kx)}{kx} = 1
\]

where \( k \) is a constant. In our case, the numerator is \( \sin(2x) \), so \( k = 2 \), and the denominator is \( 3x \).

Here are the steps:

1. Rewrite the limit as:
   \[
   \lim_{{x \to 0}} \frac{\sin(2x)}{3x} = \frac{1}{3} \lim_{{x \to 0}} \frac{\sin(2x)}{2x} \times 2
   \]

2. Apply the standard limit rule:
   \[
   \lim_{{x \to 0}} \frac{\sin(2x)}{2x} = 1
   \]

3. Therefore, the limit becomes:
   \[
   \frac{1}{3} \times 1 \times 2 = \frac{2}{3}
   \]

The solution to the limit is:

\[
\lim_{{x \to 0}} \frac{\sin(2x)}{3x} = \frac{2}{3}
\]

Let me know if you need more details or have any questions.

Here are five related questions to expand on this concept:

1. What is the limit of \( \frac{\sin(3x)}{4x} \) as \( x \to 0 \)?
2. How would you evaluate \( \lim_{{x \to 0}} \frac{\sin(x)}{5x} \)?
3. Can you use L'Hôpital's Rule to solve \( \lim_{{x \to 0}} \frac{\sin(kx)}{mx} \)?
4. What is the general form of the small angle approximation for trigonometric functions?
5. How does the limit \( \frac{\sin(x)}{x} \) behave for larger angles?

**Tip**: Always check for simplifications involving trigonometric identities before applying limit theorems.

Kerjakan soal berikut menggunakan rumus limit trigonometri: lim(x -> 0) sin(2x) / 3x

This problem involves using the trigonometric limit theorem to solve lim(x -> 0) sin(2x) / 3x. The solution walks through key concepts like trigonometric limits, small angle approximations, and applying formulas step by step. Suitable for high school calculus students.

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