https://file.aiquickdraw.com/mathbot/file/1725840753934hf9uwph0.jpg

The limit presented in the image is:

\[
\lim_{x \to 0} \frac{1 - \cos^2(2x)}{(2x)^2}
\]

We can simplify and solve this limit step-by-step.

### Step 1: Use a trigonometric identity

We know the identity \( 1 - \cos^2(\theta) = \sin^2(\theta) \). Applying this to the numerator:

\[
1 - \cos^2(2x) = \sin^2(2x)
\]

So the limit becomes:

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

### Step 2: Simplify the expression

We now rewrite the expression as:

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

### Step 3: Use the standard limit

We know from standard calculus that:

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

So, applying this to \( \frac{\sin(2x)}{2x} \):

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

### Final Answer:
\[
\lim_{x \to 0} \frac{1 - \cos^2(2x)}{(2x)^2} = 1
\]

Would you like any further details or explanations on this solution?

Here are 5 related questions to expand your understanding:

1. How does the limit of \( \frac{\sin(x)}{x} \) help in solving other trigonometric limits?
2. What other trigonometric identities can be applied to simplify limits?
3. How would the limit change if we were using a different trigonometric function, such as \( \tan(x) \)?
4. What are common standard limits used in calculus, and why are they important?
5. How do you handle indeterminate forms like \( \frac{0}{0} \) in limits?

**Tip**: Always check for trigonometric identities when dealing with limits involving trigonometric functions—they can simplify the problem significantly!

Explore how to solve the limit of (1 - cos^2(2x)) / (2x)^2 as x approaches 0, using trigonometric identities and limit properties. Learn the step-by-step approach and key formulas used, such as the identity 1 - cos^2(θ) = sin^2(θ) and the standard limit lim x→0 (sin(x)/x) = 1. Suitable for Grades 11-12.

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