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Question Number 155639 by cortano last updated on 03/Oct/21

Commented by yeti123 last updated on 03/Oct/21

$$\underset{\theta \to 0}{\lim }\frac{\sin 2\theta }{1-\cos \theta }=\underset{\theta \to 0}{\lim }\frac{\sin 2\theta }{{2\sin }^2\left(\theta /2\right)}$$$$=\underset{\theta \to 0}{\lim }\left(\frac{\sin 2\theta }{2\theta }\times \frac{{\left(\theta /2\right)}^2}{{2\sin }^2\left(\theta /2\right)}\times \frac{2\theta }{{\left(\theta /2\right)}^2}\right)$$$$=1\times \frac{1}{2}\times \underset{\theta \to 0}{\lim }\frac{8}{\theta }$$$$=\mathrm{\infty }$$

Commented by cortano last updated on 03/Oct/21

$$\mathrm{the}\mathrm{limit}{\mathrm{doesn}}^{\prime }\mathrm{t}\mathrm{exist}$$

Commented by cortano last updated on 03/Oct/21

$$\underset{x\to 0^{+}}{\lim }\frac{1}{\mathrm{x}}=\mathrm{\infty }$$$$\underset{x\to 0^{-}}{\lim }\frac{1}{\mathrm{x}}=-\mathrm{\infty }$$

Commented by yeti123 last updated on 03/Oct/21

$$=1\times \frac{1}{2}\times \underset{\theta \to 0}{\lim }\frac{8}{\theta }$$$$=\underset{\theta \to 0}{\lim }\frac{4}{\theta }$$$$=\mathrm{two}-\mathrm{sided}\mathrm{limit}\mathrm{doesnt}\mathrm{exist}$$

Answered by Ar Brandon last updated on 03/Oct/21

$$=\underset{\vartheta \to 0}{\lim }\frac{4\vartheta (1-\frac{\vartheta }{2})}{{\vartheta }^2}=\underset{x\to 0}{\lim }(\frac{4}{\vartheta }-2)\to \mathrm{infinity}$$

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