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Question Number 105106 by bemath last updated on 26/Jul/20

$$\underset{x\to 0}{\lim }\frac{\sin \left(\pi \cos {}^{2}x\right)}{3x^2}?$$

Answered by bramlex last updated on 26/Jul/20

$$\underset{x\to 0}{\lim }\frac{\sin \left(\pi \cos {}^{2}x\right)}{3x^2}=$$$$\underset{x\to 0}{\lim }\frac{-2\pi \cos x\sin x.\cos \left(\pi \cos {}^{2}x\right)}{6x}$$$$\underset{x\to 0}{\lim }\frac{-\pi \sin \left(2x\right).\cos \left(\pi \cos {}^{2}x\right)}{6x}$$$$=\underset{x\to 0}{\lim }\{-\pi \cos \left(\pi \cos {}^{2}x\right)\}.\underset{x\to 0}{\lim }\frac{\sin \left(2x\right)}{6x}$$$$=\frac{\pi }{3}▴$$

Answered by OlafThorendsen last updated on 26/Jul/20

$$\underset{x\to 0}{\lim }\frac{\sin \left(\pi {(1-\frac{x^2}{2})}^2\right)}{3x^2}$$$$\underset{x\to 0}{\lim }\frac{\sin \left(\pi (1-x^2)\right)}{3x^2}$$$$\underset{x\to 0}{\lim }\frac{\sin \left(\pi x^2\right)}{3x^2}$$$$\underset{x\to 0}{\lim }\frac{\pi }{3}.\frac{\sin \left(\pi x^2\right)}{\pi x^2}$$$$\underset{\mathrm{X}\to 0}{\lim }\frac{\pi }{3}.\frac{\mathrm{sinX}}{\mathrm{X}}=\frac{\pi }{3}$$

Answered by mathmax by abdo last updated on 26/Jul/20

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)=\frac{\sin \left(\pi {\cos }^2\mathrm{x}\right)}{{3\mathrm{x}}^2}\mathrm{we}\mathrm{have}\sin \left(\pi {\cos }^2\mathrm{x}\right)=\sin \left(\pi \times \frac{1+\cos \left(2\mathrm{x}\right)}{2}\right)$$$$=\sin (\frac{\pi }{2}+\frac{\pi }{2}\cos \left(2\mathrm{x}\right))=\cos \left(\frac{\pi }{2}\cos \left(2\mathrm{x}\right)\right)\sim \cos \left(\frac{\pi }{2}(1-{2\mathrm{x}}^2)\right)=\sin \left(\pi {\mathrm{x}}^2\right)\Rightarrow$$$$\mathrm{f}\left(\mathrm{x}\right)\sim \frac{\sin \left(\pi {\mathrm{x}}^2\right)}{{3\mathrm{x}}^2}\sim \frac{\pi {\mathrm{x}}^2}{{3\mathrm{x}}^2}=\frac{\pi }{3}\Rightarrow {\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)=\frac{\pi }{3}$$

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