Limits, Continuity & Differentiability
Trigonometric Limits
Grade 12
Question:
<p>Find the value of \(\lim_{\theta \to 0} \frac{\tan(\pi \cos^2 \theta)}{\sin(2\pi \sin^2 \theta)}\)</p>
<p>(a) \(-\frac{1}{2}\)</p>
<p>(b) \(-\frac{1}{4}\)</p>
<p>(c) \(0\)</p>
<p>(d) \(\frac{1}{4}\)</p>
Step-by-Step Solution
Key Concept: Use the approximations for small angles and the fact that \(\cos^2 \theta \approx 1 - \sin^2 \theta\) near \(\theta = 0\)
<p>As \(\theta \to 0\), \(\cos^2 \theta \to 1\) and \(\sin^2 \theta \to 0\). Use Taylor expansions: \(\tan(\pi - \pi \sin^2 \theta) \approx \tan(\pi \sin^2 \theta)\) and \(\sin(2\pi \sin^2 \theta) \approx 2\pi \sin^2 \theta\).</p>
Correct Answer: B