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Question Number 87061 by john santu last updated on 02/Apr/20

$$\underset{x\to 0}{\lim }\frac{\cos {}^3\left(2\mathrm{x}\right)-\cos \left(\mathrm{x}\right)}{\cos {}^2\left(2\mathrm{x}\right)-\cos \left(\mathrm{x}\right)}=$$

Commented by jagoll last updated on 02/Apr/20

$$\mathrm{i}\mathrm{can}\mathrm{try}$$$$\mathrm{let}\cos 2\mathrm{x}=\mathrm{t}\Rightarrow 2\cos {}^2\left(\mathrm{x}\right)-1=\mathrm{t}$$$$\cos \left(\mathrm{x}\right)=\sqrt{\frac{\mathrm{t}+1}{2}}$$$$\underset{\mathrm{t}\to 1}{\lim }\frac{{\mathrm{t}}^3-\sqrt{\frac{\mathrm{t}+1}{2}}}{{\mathrm{t}}^2-\sqrt{\frac{\mathrm{t}+1}{2}}}=$$$$\underset{\mathrm{t}\to 1}{\lim }\frac{{3\mathrm{t}}^2-\frac{1}{\sqrt{2}}\left(\frac{1}{2\sqrt{\mathrm{t}+1}}\right)}{2\mathrm{t}-\frac{1}{\sqrt{2}}\left(\frac{1}{2\sqrt{\mathrm{t}+1}}\right)}=$$$$\frac{3-\frac{1}{4}}{2-\frac{1}{4}}=\frac{11}{7}.$$

Commented by john santu last updated on 02/Apr/20

$$\mathrm{waw}..\mathrm{cool}...$$

Answered by MJS last updated on 02/Apr/20

$$\left(1\right)$$$$u\left(x\right)={\cos }^{3}2x-\cos x$$$$v\left(x\right)={\cos }^{2}2x-\cos x$$$$\underset{x\to 0}{\lim }\frac{u\left(x\right)}{v\left(x\right)}=\underset{x\to 0}{\lim }\frac{u^{\prime }\left(x\right)}{v^{\prime }\left(x\right)}=\underset{x\to 0}{\lim }\frac{u^{″}\left(x\right)}{v^{″}\left(x\right)}=\frac{11}{7}$$$$\left(2\right)$$$$\mathrm{let}x=2\arctan t;x=0\iff t=0$$$$\underset{x\to 0}{\lim }\frac{{\cos }^{3}2x-\cos x}{{\cos }^{2}2x-\cos x}=$$$$=\underset{t\to 0}{\lim }\frac{t^{10}-7t^8+58t^6-126t^4+53t^2-11}{t^{10}-3t^8+10t^6+26t^4+5t^2-7}=\frac{11}{7}$$

Commented by jagoll last updated on 02/Apr/20

$$\mathrm{waw}...\mathrm{funtastic}$$

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