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

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

Answered by Dwaipayan Shikari last updated on 26/Jul/20

lim_(x→(π/4)) ((cosx−sinx−(√2)sec^2 x)/(cosx+sinx))=(((1/(√2))−(1/(√2))−(√2).2)/(√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