Find the limit as x goes to 0 of the function sin2x/sin4x

To find the limit of the function \(\frac{\sin(2x)}{\sin(4x)}\) as \(x \to 0\), let's analyze the behavior of the function as \(x\) approaches 0.

We can use the standard limit \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\) to simplify this expression.

### Step-by-Step Solution:

1. **Rewrite the expression:**

   \[
   \lim_{x \to 0} \frac{\sin(2x)}{\sin(4x)}
   \]

2. **Multiply and divide by \(2x\) in the numerator and \(4x\) in the denominator:**

   \[
   \lim_{x \to 0} \frac{\sin(2x)}{2x} \cdot \frac{2x}{\sin(4x)} \cdot \frac{4x}{4x}
   \]

   This simplifies to:

   \[
   \lim_{x \to 0} \left( \frac{\sin(2x)}{2x} \right) \cdot \left( \frac{2x}{\sin(4x)} \right) \cdot \frac{4x}{4x}
   \]

3. **Apply the standard limit** \(\lim_{x \to 0} \frac{\sin(kx)}{kx} = 1\) for any constant \(k\):

   \[
   \lim_{x \to 0} \left( \frac{\sin(2x)}{2x} \right) = 1
   \]
   \[
   \lim_{x \to 0} \left( \frac{4x}{\sin(4x)} \right) = 1
   \]

   Thus, we have:

   \[
   \lim_{x \to 0} \left( \frac{\sin(2x)}{2x} \right) \cdot \left( \frac{4x}{\sin(4x)} \right) = 1 \cdot 1 = 1
   \]

4. **Combine the constants:**

   \[
   \lim_{x \to 0} \frac{\sin(2x)}{\sin(4x)} = \lim_{x \to 0} \left( \frac{2}{4} \cdot \frac{\sin(2x)}{2x} \cdot \frac{4x}{\sin(4x)} \right)
   \]

   Simplifying this:

   \[
   \lim_{x \to 0} \frac{\sin(2x)}{\sin(4x)} = \lim_{x \to 0} \frac{1}{2} \cdot \left( \frac{\sin(2x)}{2x} \right) \cdot \left( \frac{4x}{\sin(4x)} \right)
   \]

   Given the limits are both 1 as \(x \to 0\):

   \[
   \lim_{x \to 0} \frac{\sin(2x)}{\sin(4x)} = \frac{1}{2}
   \]

### Conclusion:

\[
\lim_{x \to 0} \frac{\sin(2x)}{\sin(4x)} = \frac{1}{2}
\]

Would you like more details on any part of this solution, or do you have any other questions?

Here are some related questions you might find interesting:

1. What is the limit of \(\frac{\sin(3x)}{\sin(5x)}\) as \(x \to 0\)?
2. How does the limit change if we replace \(\sin\) with \(\tan\) in the original function?
3. Can we use L'Hôpital's Rule to solve this limit? If so, how would it work?
4. What is the general rule for the limit \(\lim_{x \to 0} \frac{\sin(ax)}{\sin(bx)}\)?
5. What happens to the limit if \(x\) approaches a different value, like \(\pi\)?

**Tip:** Knowing the standard limits for trigonometric functions near zero can greatly simplify solving limits involving sine and cosine.

Learn how to find the limit of the function sin(2x)/sin(4x) as x approaches 0. Understand the application of trigonometric limits and basic trigonometric identities. Suitable for advanced high school students and those interested in calculus and trigonometry.

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