<p>\(\lim_{x \to \pi/4} \dfrac{\cot^3 x - \tan x}{\cos(x + \pi/4)}\) is __________.</p>
Step-by-Step Solution
Key Concept: Recognize this is a 0/0 indeterminate form at x = π/4. Rewrite the numerator using trigonometric identities to simplify, or apply L'Hôpital's rule strategically. The denominator cos(x + π/4) has a known value at x = π/4.
<p><strong>Step 1: Check the form at x = π/4</strong></p><p>At x = π/4: cot(π/4) = 1, tan(π/4) = 1, so numerator = 1³ - 1 = 0</p><p>Denominator: cos(π/4 + π/4) = cos(π/2) = 0</p><p>This is a 0/0 indeterminate form. ✓</p><p><strong>Step 2: Simplify the numerator</strong></p><p>cot³x - tan x = (cos³x/sin³x) - (sin x/cos x)</p><p>= (cos⁴x - sin⁴x)/(sin³x cos x)</p><p>= (cos²x - sin²x)(cos²x + sin²x)/(sin³x cos x)</p><p>= (cos²x - sin²x)/(sin³x cos x) [since cos²x + sin²x = 1]</p><p><strong>Step 3: Apply L'Hôpital's Rule</strong></p><p>Since we have 0/0, apply L'Hôpital's rule:</p><p>d/dx[cot³x - tan x] = -3cot²x csc²x - sec²x</p><p>d/dx[cos(x + π/4)] = -sin(x + π/4)</p><p><strong>Step 4: Evaluate at x = π/4</strong></p><p>Numerator derivative: -3(1)²(√2)² - (√2)² = -3(2) - 2 = -8</p><p>Denominator derivative: -sin(π/2) = -1</p><p><strong>Step 5: Calculate the limit</strong></p><p>lim = (-8)/(-1) = 8</p><p><strong>∴ Answer: 8</strong></p>
Correct Answer: 8