Limits, Continuity & Differentiability
Limits
Grade 12
Question:
<p>\(\lim_{x \to 0} \dfrac{x\tan 2x - 2x\tan x}{(1-\cos 2x)^2}\)</p>
<p>(1) \(\dfrac{1}{4}\)</p>
<p>(2) 1</p>
<p>(3) \(\dfrac{1}{2}\)</p>
<p>(4) \(-\dfrac{1}{2}\)</p>
Step-by-Step Solution
Key Concept: Expand tan(2x) and tan(x) using Taylor series to identify the leading term behavior, recognizing that (1-cos 2x)² has a fourth-order zero at x=0, making the denominator the critical limiting factor.
<p><strong>Step 1:</strong> Recognize the 0/0 indeterminate form. Use Taylor expansions around x=0:</p><p>tan(x) = x + x³/3 + 2x⁵/15 + ...</p><p>tan(2x) = 2x + 8x³/3 + 64x⁵/15 + ...</p><p><strong>Step 2:</strong> Find the numerator:</p><p>x·tan(2x) = x(2x + 8x³/3 + ...) = 2x² + 8x⁴/3 + ...</p><p>2x·tan(x) = 2x(x + x³/3 + ...) = 2x² + 2x⁴/3 + ...</p><p>x·tan(2x) - 2x·tan(x) = 8x⁴/3 - 2x⁴/3 + ... = 2x⁴ + O(x⁶)</p><p><strong>Step 3:</strong> Expand the denominator using cos(2x) = 1 - 2x² + 2x⁴/3 + ...</p><p>1 - cos(2x) = 2x² - 2x⁴/3 + ...</p><p>(1 - cos(2x))² = (2x² - 2x⁴/3 + ...)² = 4x⁴ - 8x⁶/3 + ... = 4x⁴ + O(x⁶)</p><p><strong>Step 4:</strong> Calculate the limit:</p><p>lim = (2x⁴)/(4x⁴) = 1/2</p><p>∴ Answer: <strong>1/2</strong></p>
Correct Answer: C