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Question Number 111923 by bemath last updated on 05/Sep/20

$$\sqrt{bemath}$$$$\underset{x\to 0}{\lim }\frac{\sqrt[3]{\cos 4x}-1}{\cos 3x-\cos 9x}?$$

Answered by bobhans last updated on 05/Sep/20

$$\underset{x\to 0}{\lim }\frac{\sqrt[3]{\cos 4\mathrm{x}}-1}{\cos 3\mathrm{x}-\cos 9\mathrm{x}}?$$$$\mathrm{recall}{\mathrm{a}}^3-{\mathrm{b}}^3=(\mathrm{a}-\mathrm{b})({\mathrm{a}}^2+{\mathrm{b}}^2+\mathrm{ab})$$$$\mathrm{then}\sqrt[3]{\cos 4\mathrm{x}}-1=\frac{\cos 4\mathrm{x}-1}{\sqrt[3]{\cos {}^{2}4\mathrm{x}}+1+\sqrt[3]{\cos 4\mathrm{x}}}$$$$\underset{x\to 0}{\lim }\frac{\cos 4\mathrm{x}-1}{\cos 3\mathrm{x}-\cos 9\mathrm{x}}.\underset{x\to 0}{\lim }\frac{1}{\sqrt[3]{\cos {}^{2}4\mathrm{x}}+1+\sqrt[3]{\cos 4\mathrm{x}}}=$$$$\underset{x\to 0}{\lim }\frac{\frac{-1}{2}.{16\mathrm{x}}^2}{-\frac{1}{2}.{9\mathrm{x}}^2+\frac{1}{2}.{81\mathrm{x}}^2}\times \frac{1}{3}=$$$$-\frac{1}{3}.\underset{x\to 0}{\lim }\frac{{16\mathrm{x}}^2}{{72\mathrm{x}}^2}=-\frac{2}{27}$$

Answered by mathmax by abdo last updated on 05/Sep/20

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)=\frac{\left({}^3\sqrt{\cos \left(4\mathrm{x}\right)}\right)-1}{\cos \left(3\mathrm{x}\right)-\cos \left(9\mathrm{x}\right)}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{(\cos {\left(4\mathrm{x}\right)}^{\frac{1}{3}}-1}{\cos \left(3\mathrm{x}\right)-\cos \left(9\mathrm{x}\right)}$$$$\mathrm{we}\mathrm{hsve}\cos \left(4\mathrm{x}\right)\sim 1-\frac{{\left(4\mathrm{x}\right)}^2}{2}=1-{8\mathrm{x}}^2(\mathrm{x}\in \mathrm{v}\left(0\right))\Rightarrow$$$${\left(\cos \left(4\mathrm{x}\right)\right)}^{\frac{1}{3}}\sim {(1-{8\mathrm{x}}^2)}^{\frac{1}{3}}\sim 1-\frac{8}{3}{\mathrm{x}}^2\mathrm{also}\cos \left(3\mathrm{x}\right)\sim 1-\frac{{9\mathrm{x}}^2}{2}$$$$\mathrm{and}\cos \left(9\mathrm{x}\right)\sim 1-\frac{{81\mathrm{x}}^2}{2}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)\sim \frac{-\frac{8}{3}{\mathrm{x}}^2}{-\frac{{9\mathrm{x}}^2}{2}+\frac{81}{2}{\mathrm{x}}^2}\Rightarrow$$$$\mathrm{f}\left(\mathrm{x}\right)\sim -\frac{8}{3.36}=-\frac{2}{27}\Rightarrow {\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)=-\frac{2}{27}$$

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