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How can I solve the following problem without the use of the L'Hopitals's rule?

$$\lim_{x\to0} \frac{1-\cos^3(x)}{x\sin{(2x)}}$$

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Hint:

We have that $1-\cos^3x = (1-\cos x)(1+\cos x + \cos^2 x)$ and $x\sin 2x = 2x\sin x \cos x$, so by limit arithmetic $$\lim_{x \to 0} \frac{1-\cos^3x}{x\sin 2x} = \lim_{x\to0} \frac{(1-\cos x)(1+\cos x + \cos^2 x)}{2x\sin x \cos x} = \frac{3}{2}\lim_{x \to 0}\frac{1-\cos x}{x\sin x}$$ From here, see if you can get a numerator in terms of $\sin x$.

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Hint: Factor $1-\cos^3 x$ and note that $1-\cos x=2\sin^2(x/2)$.

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we have $-\frac{(\cos(x)-1)(\cos(x)^2+\cos(x)+1)}{2x\sin(x)\cos(x)}=-\frac{(\cos(x)^2-1)(\cos(x)^2+\cos(x)+1)}{2x\sin(x)\cos(x)(1+\cos(x))}=\frac{(\sin(x)(\cos(x)^2+\cos(x)+1)}{2x\cos(x)(1+\cos(x))}$ and the searched limit is $\frac{3}{4}$

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$$\frac{1-\cos^3(x)}{x\sin(2x)}=\frac{(1-\cos(x))(1+\cos(x)+\cos^2(x))}{2x\sin(x)\cos(x)}$$ $$=\frac{2\sin^2(x/2)(1+\cos(x)+\cos^2(x))}{4x\sin(x/2)\cos(x/2)\cos(x)}$$ $$=\frac{1}{4}\frac{\sin(x/2)}{x/2}\frac{1+\cos(x)+\cos^2(x)}{\cos(x/2)\cos(x)}$$

The limit is 3/4.

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$$\frac{1-\cos^3(x)}{x\sin 2x}= \frac{1-(1-3x^2/2+O(x^4))}{2x^2-O(x^4)}=\frac{3x^2/2+O(x^4)}{2x^2+O(x^4)}=\frac 34$$

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Yes, you can: remember the two basic limits $$ \lim_{x\to0}\frac{\sin x}{x}=1, \qquad \lim_{x\to0}\frac{1-\cos x}{x^2}=\frac{1}{2} $$ and rewrite your limit as $$ \lim_{x\to0}\frac{1}{2}\frac{1-\cos x}{x^2}\frac{2x}{\sin2x}(1+\cos x+\cos^2x) $$

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  • $\begingroup$ The second one is not $1/2$. It is $0$. $\endgroup$ – Jack Oct 7 '16 at 3:07
  • $\begingroup$ @Jack Right. I forgot a two! $\endgroup$ – egreg Oct 7 '16 at 8:04

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