lim-x-pi-8-cot-4x-cos-4x-pi-8x-3-

Question Number 115195 by bemath last updated on 24/Sep/20

\lim_{x \to \frac{π}{8}} \frac{\cot 4 x - \cos 4 x}{{(π - 8 x)}^{3}} ?

Answered by bobhans last updated on 24/Sep/20

l e t x = \frac{π}{8} + p; p \to 0

\lim_{p \to 0} \frac{\cot (4 p + \frac{π}{2}) - \cos (4 p + \frac{π}{2})}{{(- 8 p)}^{3}} =

\lim_{p \to 0} \frac{- \tan 4 p + \sin 4 p}{- 512 p^{3}} =

\lim_{p \to 0} \frac{\tan 4 p - \sin 4 p}{512 p^{3}} = \lim_{p \to 0} \frac{\tan 4 p (1 - \cos 4 p)}{512 p^{3}}

= \frac{4 \times 8}{64 \times 8} = \frac{1}{16}

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