Ltx→π4cot3x-tanxcosx+π4=

limx→π/4 cot3⁡x−tan⁡xcos⁡(x+π/4)=limx→π/4 1tan3⁡x−tan⁡x12(cos⁡x−sin⁡x)=limx→π/4 1−tan4⁡xtan3⁡x12(cos⁡x−sin⁡x)=limx→π/4 cos4⁡x−sin4⁡xcos4⁡xsin3⁡xcos3⁡x×112(cos⁡x−sin⁡x)=limx→π/4 cos4⁡x−sin4⁡xcos⁡x⋅sin3⁡x×2(cos⁡x−sin⁡x)=limx→π/4 cos2⁡x−sin2⁡xcos2⁡x+sin2⁡xcos⁡x⋅sin3⁡x×2(cos⁡x−sin⁡x)=limx→π/4 (cos⁡x−sin⁡x)(cos⁡x+sin⁡x)⋅(1)cos⁡x⋅sin3⁡x×2cos⁡x−sin⁡x=limx→π/4 2(cos⁡x+sin⁡x)cos⁡x⋅sin3⁡x=2(cos⁡π/4+sin⁡π/4)cos⁡π/4(sin⁡π/4)3=212+1212×122=2×2(2)2=8

Ltx→π4cot3x-tanxcosx+π4=

limx→π/4 cot3⁡x−tan⁡xcos⁡(x+π/4)=limx→π/4 1tan3⁡x−tan⁡x12(cos⁡x−sin⁡x)=limx→π/4 1−tan4⁡xtan3⁡x12(cos⁡x−sin⁡x)=limx→π/4 cos4⁡x−sin4⁡xcos4⁡xsin3⁡xcos3⁡x×112(cos⁡x−sin⁡x)=limx→π/4 cos4⁡x−sin4⁡xcos⁡x⋅sin3⁡x×2(cos⁡x−sin⁡x)=limx→π/4 cos2⁡x−sin2⁡xcos2⁡x+sin2⁡xcos⁡x⋅sin3⁡x×2(cos⁡x−sin⁡x)=limx→π/4 (cos⁡x−sin⁡x)(cos⁡x+sin⁡x)⋅(1)cos⁡x⋅sin3⁡x×2cos⁡x−sin⁡x=limx→π/4 2(cos⁡x+sin⁡x)cos⁡x⋅sin3⁡x=2(cos⁡π/4+sin⁡π/4)cos⁡π/4(sin⁡π/4)3=212+1212×122=2×2(2)2=8

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$\underset{x \to \frac{π}{4}}{L t} \frac{\cot^{3} x - \tan x}{\cos (x + \frac{π}{4})} =$

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$8 \sqrt{2}$

$4 \sqrt{2}$

answer is C.

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Detailed Solution

$\begin{array}{l} lim_{x \to π / 4} \frac{\cot^{3} x - \tan x}{\cos (x + π / 4)} \\ = lim_{x \to π / 4} \frac{\frac{1}{\tan^{3} x} - \tan x}{\frac{1}{\sqrt{2}} (\cos x - \sin x)} \\ = lim_{x \to π / 4} \frac{\frac{1 - \tan^{4} x}{\tan^{3} x}}{\frac{1}{\sqrt{2}} (\cos x - \sin x)} \\ = lim_{x \to π / 4} \frac{\frac{\cos^{4} x - \sin^{4} x}{\cos^{4} x}}{\frac{\sin^{3} x}{\cos^{3} x}} \times \frac{1}{\frac{1}{\sqrt{2}} (\cos x - \sin x)} \\ = lim_{x \to π / 4} \frac{\cos^{4} x - \sin^{4} x}{\cos x \cdot \sin^{3} x} \times \frac{\sqrt{2}}{(\cos x - \sin x)} \\ = lim_{x \to π / 4} \frac{(\cos^{2} x - \sin^{2} x) (\cos^{2} x + \sin^{2} x)}{\cos x \cdot \sin^{3} x} \times \frac{\sqrt{2}}{(\cos x - \sin x)} \\ = lim_{x \to π / 4} \frac{(\cos x - \sin x) (\cos x + \sin x) \cdot (1)}{\cos x \cdot \sin^{3} x} \times \frac{\sqrt{2}}{\cos x - \sin x} \\ = lim_{x \to π / 4} \frac{\sqrt{2} (\cos x + \sin x)}{\cos x \cdot \sin^{3} x} \\ = \frac{\sqrt{2} (\cos π / 4 + \sin π / 4)}{\cos π / 4 (\sin π / 4)^{3}} = \frac{\sqrt{2} (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}})}{\frac{1}{\sqrt{2}} \times \frac{1}{2 \sqrt{2}}} \\ = 2 \times 2 (\sqrt{2})^{2} = 8 \end{array}$