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Question Number 124129 by bramlexs22 last updated on 01/Dec/20

$$\underset{x\to 1}{\lim }(1-x)\tan \left(\frac{\pi x}{2}\right)?$$

Answered by mathmax by abdo last updated on 01/Dec/20

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)=(1-\mathrm{x})\tan \left(\frac{\pi \mathrm{x}}{2}\right)\mathrm{we}\mathrm{do}\mathrm{the}\mathrm{changement}1-\mathrm{x}=\mathrm{t}\Rightarrow$$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(1-\mathrm{t})=\mathrm{t}\tan \left(\frac{\pi }{2}(1-\mathrm{t})\right)=\mathrm{t}\tan (\frac{\pi }{2}-\frac{\pi }{2}\mathrm{t})=\frac{\mathrm{t}}{\tan \left(\frac{\pi \mathrm{t}}{2}\right)}$$$$\mathrm{x}\to 1\Rightarrow \mathrm{t}\to 0\Rightarrow \mathrm{f}(1-\mathrm{t})\sim \frac{\mathrm{t}}{\frac{\pi \mathrm{t}}{2}}=\frac{2}{\pi }\Rightarrow {\lim }_{\mathrm{x}\to 1}\mathrm{f}\left(\mathrm{x}\right)=\frac{2}{\pi }$$

Answered by malwan last updated on 01/Dec/20

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

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