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Question Number 142171 by iloveisrael last updated on 27/May/21

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

Answered by iloveisrael last updated on 27/May/21

$$\underset{x\to 0}{\lim }\frac{x{\left(2\sin {}^2\left(\frac{x}{2}\right)\right)}^2}{\tan {}^{3}x(\cos {}^{3}x-1)}=$$$$\underset{x\to 0}{\lim }\frac{4\sin {}^4\left(\frac{x}{2}\right)}{\tan {}^{2}x(-2\sin {}^2\left(\frac{x}{2}\right))(\cos {}^{2}x+\cos x+1)}=$$$$\frac{1}{3}.\underset{x\to 0}{\lim }\frac{-2\sin {}^2\left(\frac{x}{2}\right)}{\tan {}^{2}x}=\frac{1}{3}.[-2\left(\frac{1}{4}\right)]$$$$=-\frac{1}{6}.$$

Answered by liberty last updated on 27/May/21

$$\underset{x\to 0}{\lim }\frac{x{(1-(1-\frac{1}{2}{x}^2))}^2}{{(x-\frac{1}{6}{x}^3)}^3-{(x+\frac{1}{3}{x}^3)}^3}$$$$=\underset{x\to 0}{\lim }\frac{\frac{1}{4}{x}^5}{x^3(1-\frac{3x^2}{6})-x^3(1+x^2)}$$$$=\frac{1}{4}\underset{x\to 0}{\lim }\frac{x^2}{(-\frac{3}{2}{x}^2)}=\frac{1}{4}\times (-\frac{2}{3})$$$$=-\frac{1}{6}\nparallel$$

Answered by som(math1967) last updated on 27/May/21

$$\underset{\mathit{x}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\frac{\mathit{x}{(1-\mathit{c}\mathit{o}\mathit{s}\mathit{x})}^2}{{\mathit{s}\mathit{i}\mathit{n}}^3\mathit{x}(1-\frac{1}{{\mathit{c}\mathit{o}\mathit{s}}^3\mathit{x}})}$$$$\underset{\mathit{x}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\frac{1}{\frac{sinx}{\mathit{x}}}\times \frac{{(1-\mathit{c}\mathit{o}\mathit{s}\mathit{x})}^2}{{\mathit{s}\mathit{i}\mathit{n}}^2\mathit{x}({\mathit{c}\mathit{o}\mathit{s}}^3\mathit{x}-1)}\times {\mathit{c}\mathit{o}\mathit{s}}^3\mathit{x}$$$$\underset{\mathit{x}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\frac{{(1-\mathit{c}\mathit{o}\mathit{s}\mathit{x})}^2{\mathit{c}\mathit{o}\mathit{s}}^3\mathit{x}}{(1-\mathit{c}\mathit{o}\mathit{s}\mathit{x})(1+\mathit{c}\mathit{o}\mathit{s}\mathit{x})(\mathit{c}\mathit{o}\mathit{s}\mathit{x}-1)({\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x}+\mathit{c}\mathit{o}\mathit{s}\mathit{x}+1)}$$$$\underset{\mathit{x}\to 0}{\mathit{l}\mathit{i}\mathit{m}}\frac{-{\mathit{c}\mathit{o}\mathit{s}}^3\mathit{x}}{(1+\mathit{c}\mathit{o}\mathit{s}\mathit{x})({\mathit{c}\mathit{o}\mathit{s}}^2\mathit{x}+\mathit{c}\mathit{o}\mathit{s}\mathit{x}+1)}=\frac{-1}{(1+1)(1+1+1)}$$$$=-\frac{1}{6}$$

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