lim-x-pi-2n-sin-2nx-1-cos-nx-4n-2-x-2-pi-2-

Question Number 198572 by cortano12 last updated on 22/Oct/23

\lim_{x \to \frac{π}{2 n}} \frac{\sqrt{\frac{\sin 2 nx}{1 + \cos nx}}}{{4 n}^{2} x^{2} - π^{2}} = ?

Answered by MM42 last updated on 22/Oct/23

\frac{π}{2 n} - x = u

{l i m}_{u \to 0} \frac{\sqrt{\frac{s i n 2 n (\frac{π}{2 n} - u)}{1 + c o s n (\frac{π}{2 n} - u)}}}{(2 n (\frac{π}{2 n} - u) - π) (2 n (\frac{π}{2 n} - u) + π)}

= {l i m}_{u \to 0} \frac{\sqrt{\frac{s i n (2 n u)}{1 + s i n (n u)}}}{(- 2 n u) (2 π - 2 n u)}

= {l i m}_{u \to 0} \frac{\sqrt{\frac{s i n (2 n u)}{1 + s i n (n u)}}}{(- 2 n u) (2 π - 2 n u)} = {\begin{cases} + \infty i f u \to 0^{-} \\ - \infty i f u \to 0^{+} \end{cases}

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