lim-x-1-2-cot-pix-2x-2-x-1-

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

\lim_{x \to 1 / 2} (\frac{\cot (π x)}{2 x^{2} + x - 1}) = ?

Answered by liberty last updated on 18/Nov/20

\lim_{x \to 1 / 2} \frac{\cot (π x)}{(2 x - 1) (x + 1)} = \lim_{x \to 1 / 2} \frac{1}{x + 1} . \lim_{(\frac{2 x - 1}{2}) \to 0} \frac{\cot (π x)}{(2 x - 1)}

= \frac{2}{3} \times \lim_{X \to 0} \frac{\cot (π (\frac{2 X + 1}{2}))}{2 X}; [let \frac{2 x - 1}{2} = X]

= \frac{1}{3} \times \lim_{X \to 0} \frac{\cot (\frac{π}{2} + π X)}{X}

= \frac{1}{3} \times \lim_{X \to 0} \frac{- \tan (π X)}{X} = - \frac{π}{3} . ▴

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

\lim_{x \to \frac{1}{2}} \frac{c o s (π x)}{s i n (π x) (2 x - 1) (x + 1)}

\lim_{x \to \frac{1}{2}} \frac{2}{3} \frac{s i n (\frac{π}{2} - π x)}{(2 x - 1)} = \frac{2}{3} . \frac{π (1 - 2 x)}{2 (2 x - 1)} = - \frac{π}{3}