lim_(x→(π/4)) ((sin x+cos x−(√2)tan x)/(sin x−cos x))


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Question Number 105126 by bemath last updated on 26/Jul/20

$\lim_{x \to \frac{π}{4}} \frac{\sin x + \cos x - \sqrt{2} \tan x}{\sin x - \cos x}$
	Answered by Dwaipayan Shikari last updated on 26/Jul/20

	$\lim_{x \to \frac{π}{4}} \frac{cosx - sinx - \sqrt{2} \sec^{2} x}{cosx + sinx} = \frac{\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} - \sqrt{2} . 2}{\sqrt{2}} = - 2$
	Answered by bramlex last updated on 26/Jul/20
	$sin x+cos x = (√2) sin (x+(π/4)) sin x−cos x = (√2) sin (x−(π/4)) set x = w+(π/4) lim_(w→0) (((√2) sin (w+(π/2))−(√2) tan (w+(π/4)))/((√2) sin (w))) lim_(w→0) ((cos w−{((1+tan w)/(1−tan w))})/(sin w)) lim_(w→0) ((cos w−sin w−1−tan w)/(sin w(1−tan w))) lim_(x→0) (((cos w−1)−sin w−tan w)/(sin w(1−tan w))) lim_(w→0) ((((1−(w^2 /2))−1)−(w−(w^3 /6))−(w+(w^3 /3)))/((w−(w^3 /6))(1−(w+(w^3 /3))))) lim_(w→0) ((−(w^2 /2)−2w−(w^3 /6))/((w−(w^3 /6))(1−w−(w^3 /3)))) = −2$
	$\sin x + \cos x = \sqrt{2} \sin (x + \frac{π}{4})$ $\sin x - \cos x = \sqrt{2} \sin (x - \frac{π}{4})$ $s e t x = w + \frac{π}{4}$ $\lim_{w \to 0} \frac{\sqrt{2} \sin (w + \frac{π}{2}) - \sqrt{2} \tan (w + \frac{π}{4})}{\sqrt{2} \sin (w)}$ $\lim_{w \to 0} \frac{\cos w - {\frac{1 + \tan w}{1 - \tan w}}}{\sin w}$ $\lim_{w \to 0} \frac{\cos w - \sin w - 1 - \tan w}{\sin w (1 - \tan w)}$ $\lim_{x \to 0} \frac{(\cos w - 1) - \sin w - \tan w}{\sin w (1 - \tan w)}$ $\lim_{w \to 0} \frac{((1 - \frac{w^{2}}{2}) - 1) - (w - \frac{w^{3}}{6}) - (w + \frac{w^{3}}{3})}{(w - \frac{w^{3}}{6}) (1 - (w + \frac{w^{3}}{3}))}$ $\lim_{w \to 0} \frac{- \frac{w^{2}}{2} - 2 w - \frac{w^{3}}{6}}{(w - \frac{w^{3}}{6}) (1 - w - \frac{w^{3}}{3})} = - 2$
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