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Question Number 179366 by mathlove last updated on 28/Oct/22

$$\underset{x\to \pi }{\lim }\frac{sin\frac{x}{2}-1}{x-\pi }=?$$

Commented by mahdipoor last updated on 28/Oct/22

$$=D_x{\left(sin\frac{x}{2}\right)}_{x=\pi }=\frac{1}{2}cos{\left(\frac{x}{2}\right)}_{x=\pi }=0$$

Commented by mathlove last updated on 28/Oct/22

$$discraibanswer??$$

Commented by mr W last updated on 29/Oct/22

$$\underset{x\to \pi }{\lim }\frac{sin\frac{x}{2}-1}{x-\pi }$$$$=\underset{x\to \pi }{\lim }\frac{sin\frac{x}{2}-\sin \frac{\pi }{2}}{x-\pi }\leftarrow derivativeof\sin \frac{x}{2}$$$$={\left(\sin \frac{x}{2}\right)}^{\prime }{\mid }_{x=\pi }$$$$=\left(\frac{1}{2}\cos \frac{x}{2}\right){\mid }_{x=\pi }$$$$=\frac{1}{2}\cos \frac{\pi }{2}$$$$=0$$

Commented by mathlove last updated on 29/Oct/22

$$tanks$$

Commented by CElcedricjunior last updated on 29/Oct/22

$$\underset{x\to \mathit{\pi }}{\lim }\frac{\mathbf{sin}\frac{\mathbf{x}}{2}-1}{\mathbf{x}-\mathit{\pi }}=\frac{0}{0}=\mathbf{FI}$$$$\mathbf{to}\mathbf{apply}\mathbf{hospital}$$$$\iff \underset{x\to \mathit{\pi }}{\lim }\frac{\mathbf{sin}\frac{\mathbf{x}}{2}-1}{\mathbf{x}-\mathit{\pi }}=\underset{x\to \mathit{\pi }}{\lim }\frac{\frac{1}{2}\mathbf{cos}\frac{\mathbf{x}}{2}}{1}=0$$$$\underset{x\to \mathit{\pi }}{\lim }\frac{\mathbf{sin}\frac{\mathbf{x}}{2}-1}{\mathbf{x}-\mathit{\pi }}=0$$$$...........lecelebrecedricjinior......$$

Answered by a.lgnaoui last updated on 28/Oct/22

$${\lim }_{\mathrm{x}\to \pi }\frac{1}{2}\times \left(\frac{\sin \frac{x}{2}-\sin \frac{\pi }{2}}{\frac{x}{2}-\frac{\pi }{2}}\right)=\frac{1}{2}\times \frac{d{\left(\sin \frac{x}{2}\right)}_{x=\pi /2}}{dx}=\frac{1}{4}\cos \left(\frac{\pi }{4}\right)=\frac{1}{4}\times \frac{\sqrt{2}}{2}=\frac{\sqrt{2}}{8}$$

Commented by Acem last updated on 29/Oct/22

$$fix{\left(\sin \theta \right)}^{\prime }$$

Answered by cortano1 last updated on 29/Oct/22

$$\underset{x\to \pi }{\lim }\frac{\sin \frac{\mathrm{x}}{2}-1}{\mathrm{x}-\pi }$$$$\mathrm{let}\mathrm{x}-\pi =\mathrm{y}\Rightarrow \mathrm{x}=\pi +\mathrm{y}$$$$\mathrm{L}=\underset{\mathrm{y}\to 0}{\lim }\frac{\sin (\frac{\pi }{2}+\frac{\mathrm{y}}{2})-1}{\mathrm{y}}$$$$=\underset{\mathrm{y}\to 0}{\lim }\frac{\cos \frac{\mathrm{y}}{2}-1}{\mathrm{y}}$$$$=\underset{\mathrm{y}\to 0}{\lim }\frac{-\sin {}^2\left(\frac{\mathrm{y}}{2}\right)}{(\cos \frac{\mathrm{y}}{2}+1)\mathrm{y}}$$$$=-\underset{\mathrm{y}\to 0}{\lim }\frac{\sin \left(\frac{\mathrm{y}}{2}\right)}{\mathrm{y}}.\underset{\mathrm{y}\to 0}{\lim }\frac{\sin \left(\frac{\mathrm{y}}{2}\right)}{\cos \left(\frac{\mathrm{y}}{2}\right)+1}$$$$=-\frac{1}{2}\times 0=0$$

Commented by mathlove last updated on 29/Oct/22

$$thanks$$

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