lim-x-pi-4-x-pi-4-sin-3x-3-pi-4-2-1-sin2x-

Question Number 10374 by ridwan balatif last updated on 06/Feb/17

\lim_{x \to \frac{π}{4}} \frac{(x - \frac{π}{4}) \sin (3 x - 3 \frac{π}{4})}{2 (1 - \sin 2 x)} = \dots ?

Answered by mrW1 last updated on 06/Feb/17

l e t u = x - \frac{π}{4}

w i t h x \to \frac{π}{4}, u \to 0

\sin 2 x = \sin (2 u + \frac{π}{2}) = \cos (2 u) = 1 - 2 \sin^{2} u

L = \lim_{x \to \frac{π}{4}} \frac{(x - \frac{π}{4}) \sin (3 x - 3 \frac{π}{4})}{2 (1 - \sin 2 x)}

= \lim_{u \to 0} \frac{u \sin (3 u)}{4 \sin^{2} u} = \lim_{u \to 0} \frac{u \times (\frac{\sin 3 u}{3 u}) \times 3 u}{4 \times {(\frac{\sin u}{u})}^{2} \times u^{2}}

= \lim_{u \to 0} \frac{3 \times (\frac{\sin 3 u}{3 u})}{4 \times {(\frac{\sin u}{u})}^{2}} = \frac{3 \times 1}{4 \times 1^{2}} = \frac{3}{4}

Commented by ridwan balatif last updated on 06/Feb/17

thank you sir

Answered by arge last updated on 08/Feb/17

p o r l^{'} h o p i t a l,

y = \frac{(x - \frac{x}{4}) s e n (3 x - \frac{12 + π}{4})}{2 (1 - s e n 2 x)}

y^{'} = \frac{2 (1 - s e n 2 x) [(x - \frac{x}{4}) c o s (3 x - \frac{12 + π}{4}) + s e n (3 x - \frac{12 + π}{4}) (1 - \frac{1}{4})] - (x - \frac{x}{4}) s e n (3 x - \frac{12 + π}{4}) \times 2 (- c o s 2 x)}{4 (1 - 2 s e n 2 x + 4 {s e n}^{2} {x c o s}^{2} x)}

y^{'} = \frac{A}{0}

A \underset{}{=} - 134.65, y^{'} = \infty ∵∵∵∵∵ R t a

Answered by bahmanfeshki last updated on 02/Mar/17

1 - \sin 2 x = {(\sin x + \cos x)}^{2} = {2 \sin}^{2} (x + \frac{π}{4})

x + \frac{π}{4} = t

\lim_{t \to 0} \frac{t \sin 3 t}{4 \sin^{2} t} = \frac{3}{4} \lim_{t \to 0} \frac{\frac{\sin 3 t}{3 t}}{{(\frac{\sin t}{t})}^{2}} = \frac{3}{4}