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Question Number 115320 by john santu last updated on 25/Sep/20

$$\left(1\right)\underset{x\to 0}{\lim }\frac{1-\cos {}^6\left(2x\right)\cos {}^3\left(3x\right)}{3x^2}?$$$$\left(2\right)\underset{x\to 0}{\lim }\frac{1-\cos 4x+2\sin {}^{2}x.\cos 4x}{x^2.\cos 3x}?$$$$\left(3\right)\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin x-2\cos {}^{2}x-1}{\sqrt{\sin {}^{3}x}-\sqrt{\sin x}}?$$

Answered by bobhans last updated on 29/Sep/20

$$\left(1\right)\underset{x\to 0}{\lim }\frac{1-{(1-2\sin {}^{2}x)}^6{(1-2\sin {}^2\left(\frac{3}{2}x\right))}^3}{3x^2}=$$$$\underset{x\to 0}{\lim }\frac{1-(1-12x^2)(1-6.\frac{9}{4}{x}^2)}{3x^2}=$$$$\underset{x\to 0}{\lim }\frac{1-(1-12x^2)(1-\frac{27x^2}{2})}{3x^2}=$$$$\underset{x\to 0}{\lim }\frac{1-(1-\frac{51}{2}{x}^2+142x^4)}{3x^2}=\frac{17}{2}$$

Answered by bobhans last updated on 25/Sep/20

$$\left(3\right)\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin x-2\cos {}^{2}x-1}{\sqrt{\sin {}^{3}x}-\sqrt{\sin x}}$$$$settingx=\frac{\pi }{2}+w;w\to 0$$$$\underset{w\to 0}{\lim }\frac{\cos w-2\sin {}^{2}w-1}{\sqrt{\cos w}(\cos w-1)}=$$$$\underset{w\to 0}{\lim }\frac{1-\frac{1}{2}{w}^2-2w^2-1}{1.(1-\frac{1}{2}{w}^2-1)}=$$$$\underset{w\to 0}{\lim }\frac{-\frac{5}{2}{w}^2}{-\frac{1}{2}{w}^2}=5$$

Answered by bemath last updated on 25/Sep/20

$$\left(2\right)\underset{x\to 0}{\lim }\frac{1-\cos 4x+2\sin {}^{2}x.\cos 4x}{x^2.\cos 3x}=$$$$\underset{x\to 0}{\lim }\frac{1-(1-2\sin {}^{2}2x)+2\sin {}^{2}x.\cos 4x}{x^2.\cos 3x}=$$$$\underset{x\to 0}{\lim }\frac{2\sin {}^{2}2x+2\sin {}^{2}x.\cos 4x}{x^2.\cos 3x}=$$$$\underset{x\to 0}{\lim }\frac{8\sin {}^{2}x.\cos {}^{2}x+{2\sin }^{2}x.\cos 4x}{x^2.\cos 3x}=$$$$\underset{x\to 0}{\lim }\frac{2\sin {}^{2}x}{x^2}\times \underset{x\to 0}{\lim }\frac{4\cos {}^{2}x+\cos 4x}{\cos 3x}=$$$$2\times 5=10$$

Answered by Dwaipayan Shikari last updated on 25/Sep/20

$$\underset{x\to \frac{\pi }{2}}{\lim }\frac{\mathrm{sinx}-2+{2\sin }^2\mathrm{x}-1}{{\sin }^3\mathrm{x}-\mathrm{sinx}}(\sqrt{{\sin }^3\mathrm{x}}+\sqrt{\mathrm{sinx}})$$$$\underset{x\to \frac{\pi }{2}}{\lim }\frac{{2\sin }^2\mathrm{x}+\mathrm{sinx}-3}{\mathrm{sinx}(-{\cos }^2\mathrm{x})}.2$$$$\underset{x\to \frac{\pi }{2}}{\lim 2}\frac{\mathrm{sinx}(2\mathrm{sinx}+3)-(2\mathrm{sinx}+3)}{\mathrm{sinx}(-{\cos }^2\mathrm{x})}$$$$\underset{x\to \frac{\pi }{2}}{\lim 2}\frac{(\mathrm{sinx}-1)(2\mathrm{sinx}+3)}{-\mathrm{sinx}\left({\cos }^2\mathrm{x}\right)}=\underset{x\to \frac{\pi }{2}}{\lim 10}.\frac{1-\mathrm{sinx}}{\mathrm{sinx}\left({\cos }^2\mathrm{x}\right)}$$$$\underset{x\to \frac{\pi }{2}}{\lim 10}.\frac{{\cos }^2\mathrm{x}}{{\cos }^2\mathrm{xsinx}(\mathrm{sinx}+1)}=5$$

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