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Question Number 199597 by mnjuly1970 last updated on 05/Nov/23

calculate \dots

Q : \lim_{x \to \frac{π}{4}} {(1 + s i n (x) - c o s (x))}^{t a n (2 x)} = ?

∙ N i c e - M a t h e m a t i c s ∙

Answered by witcher3 last updated on 05/Nov/23

e^{tg (2 x) \ln (1 + \sin (x) - \cos (x))}

\ln (1 + \sin (x) - \cos (x))

x = \frac{π}{4} + y

tg (2 x) = - \cot (2 y)

1 + \sin (x) - \cos (x) = 1 + \frac{\sin (y) + \cos (y)}{\sqrt{2}} - \frac{\cos (y) - \sin (y)}{\sqrt{2}}

. 1 + \sqrt{2} \sin (y) = 1 + y \sqrt{2} + o (y)

\ln (1 + \sin (y) \sqrt{2}) = y \sqrt{2} + o (y {

- \cot (2 y) \ln (1 + y \sqrt{2)} {\sim - \frac{1}{\sqrt{2}}

= e^{- \frac{1}{\sqrt{2}}}

Commented by mnjuly1970 last updated on 05/Nov/23

Commented by witcher3 last updated on 06/Nov/23

thank You Withe pleasur

Answered by MM42 last updated on 05/Nov/23

{l i m}_{x \to \frac{π}{4}} e^{(s i n x - c o s x) \times t a n 2 x}

= {l i m}_{x \to \frac{π}{4}} e^{\frac{- c o s 2 x}{(s i n x + c o s x)} \times \frac{s i n 2 x}{c o s 2 x}}

= e^{- \frac{\sqrt{2}}{2}} ✓

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