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Question Number 122548 by bramlexs22 last updated on 18/Nov/20

$$\underset{x\to 1/2}{\lim }\left(\frac{\cot \left(\pi x\right)}{2x^2+x-1}\right)=?$$

Answered by liberty last updated on 18/Nov/20

$$\underset{x\to 1/2}{\lim }\frac{\cot \left(\pi \mathrm{x}\right)}{(2\mathrm{x}-1)(\mathrm{x}+1)}=\underset{x\to 1/2}{\lim }\frac{1}{\mathrm{x}+1}.\underset{\left(\frac{2\mathrm{x}-1}{2}\right)\to 0}{\lim }\frac{\cot \left(\pi \mathrm{x}\right)}{(2\mathrm{x}-1)}$$$$=\frac{2}{3}\times \underset{\mathrm{X}\to 0}{\lim }\frac{\cot \left(\pi \left(\frac{2\mathrm{X}+1}{2}\right)\right)}{2\mathrm{X}};[\mathrm{let}\frac{2\mathrm{x}-1}{2}=\mathrm{X}]$$$$=\frac{1}{3}\times \underset{\mathrm{X}\to 0}{\lim }\frac{\cot (\frac{\pi }{2}+\pi \mathrm{X})}{\mathrm{X}}$$$$=\frac{1}{3}\times \underset{\mathrm{X}\to 0}{\lim }\frac{-\tan \left(\pi \mathrm{X}\right)}{\mathrm{X}}=-\frac{\pi }{3}.▴$$

Answered by Dwaipayan Shikari last updated on 18/Nov/20

$$\underset{x\to \frac{1}{2}}{\lim }\frac{cos\left(\pi x\right)}{sin\left(\pi x\right)(2x-1)(x+1)}$$$$\underset{x\to \frac{1}{2}}{\lim }\frac{2}{3}\frac{sin(\frac{\pi }{2}-\pi x)}{(2x-1)}=\frac{2}{3}.\frac{\pi (1-2x)}{2(2x-1)}=-\frac{\pi }{3}$$

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