Limits, Continuity & Differentiability
Limits
Grade 12
Question:
<p>Evaluate: \(\lim_{x \to 0} \dfrac{(1-\cos x) + (1-\cos 2x) + \cdots + (1-\cos 10x)}{x^2}\)</p>
<p>(a) \(\dfrac{381}{2}\)</p>
<p>(b) \(190\)</p>
<p>(c) \(\dfrac{383}{2}\)</p>
<p>(d) \(\dfrac{385}{2}\)</p>
Step-by-Step Solution
Key Concept: Use the Taylor expansion cos(nx) ≈ 1 - (nx)²/2 for small x, so (1 - cos(nx)) ≈ (nx)²/2. Sum all 10 terms and divide by x² to find the limit.
<p><strong>Step 1:</strong> Recognize the 0/0 indeterminate form as x → 0. Use Taylor expansion: cos(nx) = 1 - (nx)²/2 + O(x⁴)</p><p><strong>Step 2:</strong> Therefore (1 - cos(nx)) = (nx)²/2 - O(x⁴) = n²x²/2 + O(x⁴)</p><p><strong>Step 3:</strong> Sum all terms from n=1 to n=10:</p><p>Σ(1 - cos(nx)) = (x²/2)Σn² = (x²/2)[1² + 2² + 3² + ... + 10²]</p><p><strong>Step 4:</strong> Use the formula Σn² = n(n+1)(2n+1)/6. For n=10: Σn² = 10(11)(21)/6 = 385</p><p><strong>Step 5:</strong> Divide by x²:</p><p>lim(x→0) [(x²/2)·385]/x² = 385/2 = <strong>192.5</strong></p><p>∴ Answer: D</p>
Correct Answer: D