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Question Number 159613 by cortano last updated on 19/Nov/21

Commented by tounghoungko last updated on 19/Nov/21

$$A=\underset{x\to 0}{\lim }\frac{2\sin x(1-\cos x)}{\sin x(1-\cos {}^{3}x)}$$$$A=\underset{x\to 0}{\lim }\frac{2(1-\cos x)}{(1-\cos x)(\cos {}^{2}x+\cos x+1)}$$$$A=\frac{2}{3}$$

Commented by tounghoungko last updated on 19/Nov/21

$$D=\underset{x\to \frac{\pi }{6}}{\lim }\frac{(2\sin x-1)(\sin x-1)}{\cos x(1-2\sin x)}$$$$D=\frac{\frac{1}{2}-1}{-\frac{\sqrt{3}}{2}}=\frac{1}{2}\times \frac{2}{\sqrt{3}}=\frac{1}{\sqrt{3}}$$

Commented by tounghoungko last updated on 19/Nov/21

$$E=\underset{x\to \frac{\pi }{3}}{\lim }\frac{2\cos x-1}{3-4(1-\cos {}^{2}x)}$$$$E=\underset{x\to \frac{\pi }{3}}{\lim }\frac{2\cos x-1}{(2\cos x-1)(2\cos x+1)}$$$$E=\frac{1}{2\left(\frac{1}{2}\right)+1}=\frac{1}{2}$$

Commented by tounghoungko last updated on 19/Nov/21

$$C=\underset{x\to \frac{\pi }{4}}{\lim }\frac{(\cos x-\sin x)(1+\frac{1}{2}\sin 2x)}{\cos {}^{3}x(\cos x-\sin x)}$$$$C=\frac{1+\frac{1}{2}}{{\left(\frac{\sqrt{2}}{2}\right)}^3}=\frac{3}{2}\times \frac{8}{2\sqrt{2}}=\frac{6}{\sqrt{2}}=3\sqrt{2}$$

Commented by tounghoungko last updated on 19/Nov/21

$$B=\underset{x\to \frac{\pi }{6}}{\lim }\frac{(2\sin x-1)(\sin x-1)}{2\sin x-1}$$$$B=\frac{1}{2}-1=-\frac{1}{2}$$

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