Question Number 104348 by bemath last updated on 21/Jul/20

lim_(x→0) (((arc tan (x)−arc sin (x))/(x(1−cos (x)))))

Answered by john santu last updated on 21/Jul/20

L′Hopital rule   lim_(x→0) [ ((tan^(−1) (x)−sin^(−1) (x))/(2x sin ^2 ((x/2)))) ]=  lim_(x→0)  [(((1/(1+x^2 )) −(1/(√(1−x^2 ))))/(2sin ^2 ((x/2))+xsin x)) ]=  lim_(x→0) [(((1+x^2 )^(−1) −(1−x^2 )^(−1/2) )/((x^2 /2)+x^2 )) ]=  lim_(x→0)  [(((1−x^2 )−(1+(x^2 /2)))/((3/2)x^2 ))] =  lim_(x→0)  [((−((3x^2 )/2))/((3/2)x^2 )) ] = −1  (JS ⊛ )

Answered by OlafThorendsen last updated on 21/Jul/20

arctanx ∼_0  x−(x^3 /3)  arcsinx ∼_0  x+(x^3 /6)  cosx ∼_0  1−(x^2 /2)  lim_(x→0) ((((x−(x^3 /3))−(x+(x^3 /6)))/(x(1−1+(x^2 /2)))))  lim_(x→0) (((−(x^3 /2))/(x^3 /2))) = −1

Answered by Dwaipayan Shikari last updated on 21/Jul/20

lim_(x→0) ((tan^(−1) x−sin^(−1) x)/(x(1−cosx)))=lim_(x→0) ((x−(x^3 /3)−x−(x^3 /6))/(x(1−1+(x^2 /2))))=((−(x^3 /2))/(x((x^2 /2))))=−1