Find the limit as x goes to 0 of the function sin2x/sin4x using the double angle identity

To find the limit of the function \(\lim_{{x \to 0}} \frac{\sin(2x)}{\sin(4x)}\) using the double angle identity, let's start by applying the double angle identities for sine:

\[
\sin(2x) = 2\sin(x)\cos(x)
\]
\[
\sin(4x) = 2\sin(2x)\cos(2x)
\]

Now, substitute \(\sin(2x)\) in the denominator:

\[
\frac{\sin(2x)}{\sin(4x)} = \frac{2\sin(x)\cos(x)}{2\sin(2x)\cos(2x)}
\]

Simplifying by canceling the common factor of 2:

\[
\frac{\sin(2x)}{\sin(4x)} = \frac{\sin(x)\cos(x)}{\sin(2x)\cos(2x)}
\]

Now, substitute the expression \(\sin(2x)\) again:

\[
\frac{\sin(x)\cos(x)}{2\sin(x)\cos(x)\cos(2x)} = \frac{1}{2\cos(2x)}
\]

Taking the limit as \(x \to 0\):

\[
\lim_{{x \to 0}} \frac{1}{2\cos(2x)} = \frac{1}{2\cos(0)} = \frac{1}{2 \times 1} = \frac{1}{2}
\]

So, the limit is:

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

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the limit of \(\lim_{{x \to 0}} \frac{\sin(3x)}{\sin(5x)}\)?
2. How does L'Hôpital's Rule apply to trigonometric limits?
3. What is the Taylor series expansion for \(\sin(x)\) near \(x = 0\)?
4. How do you evaluate limits involving sine and cosine functions using their small-angle approximations?
5. What is the limit \(\lim_{{x \to 0}} \frac{1 - \cos(x)}{x^2}\) and how is it derived?

**Tip:** For small angles, the sine function can be approximated by \(\sin(x) \approx x\), which is useful in limit problems.

Learn how to find the limit of the function lim(x → 0) sin(2x)/sin(4x) using the double angle identity for sine. Detailed step-by-step solution with application of trigonometric identities. Suitable for college-level students and calculus enthusiasts.

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