Question

\( \lim_{x \to 0} \frac{\sin^2(\pi \cos^4 x)}{x^4} \) is equal to :

• A. \( 2\pi^2 \)
• B. \( \pi^2 \)
• C. \( 4\pi^2 \)
• D. \( 4\pi \)

Hint:

When you see \( \sin(f(x)) \) where \( f(x) \to \pi \) as \( x \to 0 \), use the identity \( \sin(\pi - \theta) \) to bring the argument to zero so you can apply the \( \frac{\sin \theta}{\theta} \) rule.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
As \( x \to 0 \), the argument of the sine function approaches \( \pi \). We use trigonometric identities to transform the argument into something that approaches \( 0 \) and then apply the standard limit \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \).
Step 2: Detailed Explanation:
1. Since \( \sin^2(\theta) = \sin^2(\pi - \theta) \), we can write:
\[ \sin^2(\pi \cos^4 x) = \sin^2(\pi - \pi \cos^4 x) = \sin^2(\pi(1 - \cos^4 x)) \]
2. Factorize \( 1 - \cos^4 x \):
\[ 1 - \cos^4 x = (1 - \cos^2 x)(1 + \cos^2 x) = \sin^2 x (1 + \cos^2 x) \]
3. The expression becomes:
\[ \lim_{x \to 0} \frac{\sin^2(\pi \sin^2 x (1 + \cos^2 x))}{x^4} \]
4. Multiply and divide by the squared interior term to use the standard limit:
\[ \lim_{x \to 0} \left[ \frac{\sin(\pi \sin^2 x (1 + \cos^2 x))}{\pi \sin^2 x (1 + \cos^2 x)} \right]^2 \cdot \frac{\pi^2 \sin^4 x (1 + \cos^2 x)^2}{x^4} \]
5. Evaluate the limits:
- The term in the square brackets goes to \( 1^2 = 1 \).
- The remaining part is \( \pi^2 \cdot \left( \lim_{x \to 0} \frac{\sin x}{x} \right)^4 \cdot (1 + \cos^2 0)^2 \).
- This is \( \pi^2 \cdot 1^4 \cdot (1 + 1)^2 = \pi^2 \cdot 4 = 4\pi^2 \).
Step 3: Final Answer:
The limit is \( 4\pi^2 \).