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Question Number 112782 by bemath last updated on 09/Sep/20

$$\mathrm{without}{\mathrm{L}}^{\prime }\mathrm{Hopital}$$$$\underset{x\to \pi }{\lim }\frac{\cos \left(\frac{\mathrm{x}}{2}\right)}{\mathrm{x}-\pi }$$

Answered by john santu last updated on 09/Sep/20

$$settingx=\pi +s,s\to 0$$$$\underset{s\to 0}{\lim }\frac{\cos (\frac{\pi }{2}+\frac{s}{2})}{s}=\underset{s\to 0}{\lim }\frac{-\sin \left(\frac{s}{2}\right)}{s}=-\frac{1}{2}$$

Answered by Dwaipayan Shikari last updated on 09/Sep/20

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

Answered by Aziztisffola last updated on 09/Sep/20

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

Commented by malwan last updated on 09/Sep/20

$$thisis{L}^{{}^{\prime }}Hopital$$

Commented by Aziztisffola last updated on 09/Sep/20

$$\mathrm{I}\mathrm{use}\mathrm{this}\mathrm{property};\mathrm{derivitive}\mathrm{in}\mathrm{one}$$$$\mathrm{point}{x}_0;\mathrm{this}\mathrm{is}\mathrm{not}{\mathrm{l}}^{\prime }\mathrm{Hopital}\mathrm{rule}$$$$\underset{x\to x_0}{\lim }\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}=f{}^{\prime }\left(x_0\right)$$$$\mathrm{let}f\left(x\right)=\cos \left(\frac{x}{2}\right)and{x}_0=\pi$$$$\underset{x\to \pi }{\lim }\frac{f\left(x\right)-f\left(\pi \right)}{x-\pi }=f{}^{\prime }\left(\pi \right)={\cos }^{\prime }\left(\frac{\pi }{2}\right)$$

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