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Question: Let f(x) = \(\frac{2\sqrt{2} - (\cos(x) + \sin(x))^{3}}{1 - \sin(2x)}\) then \(\underset{x \rightarr...

Let f(x) = 22−(cos⁡(x)+sin⁡(x))31−sin⁡(2x)\frac{2\sqrt{2} - (\cos(x) + \sin(x))^{3}}{1 - \sin(2x)} then Limx→π4\underset{x \rightarrow \frac{\pi}{4}}{Lim}f(x) is equal to

A

12\frac{1}{\sqrt{2}}

B

2\sqrt{2}

C

1

D

32\frac{3}{\sqrt{2}}

Answer

32\frac{3}{\sqrt{2}}

Explanation

Solution

f(x) = 22−(cos⁡x+sin⁡x)31−sin⁡2x\frac{2\sqrt{2} - (\cos x + \sin x)^{3}}{1 - \sin 2x} (0/0)

L'Hospital Rule

lim⁡x→π4\lim_{x \rightarrow \frac{\pi}{4}} f(x)

= lim⁡x→π4\lim_{x \rightarrow \frac{\pi}{4}} – 3(cos⁡x+sin⁡x)2(−sin⁡x+cos⁡x)−2cos⁡2x\frac{3(\cos x + \sin x)^{2}( - \sin x + \cos x)}{- 2\cos 2x}

= lim⁡x→π4\lim_{x \rightarrow \frac{\pi}{4}} +3(cos⁡x+sin⁡x)2(cos⁡x−sin⁡x)2(cos⁡x−sin⁡x)(cos⁡x+sin⁡x)\frac{+ 3(\cos x + \sin x)^{2}(\cos x - \sin x)}{2(\cos x - \sin x)(\cos x + \sin x)}

= lim⁡x→π4\lim_{x \rightarrow \frac{\pi}{4}} 32\frac{3}{2} × (cos x + sin x)

= 32×22\frac{3}{2} \times \frac{2}{\sqrt{2}} = 32\frac{3}{\sqrt{2}}