Question Number 115162 by john santu last updated on 24/Sep/20

 (1)  lim_(x→1)  arc sin (((1−(√x))/(1−x))) =?  (2) lim_(x→∞)  e^(x^3 +(√(cos ((1/x^2 )))))  =?  (3) lim_(x→0)  csc x .sin (sin x) =?

Answered by bobhans last updated on 24/Sep/20

(1) lim_(x→1)  arc sin (((1−(√x))/(1−x))) = arc sin (lim_(x→1) ((1−(√x))/(1−x)))                 = arc sin (lim_(x→1)  ((1−(√x))/((1−(√x))(1+(√x)))))                = arc sin (lim_(x→1)  (1/(1+(√x))))               = arc sin ((1/2)) = (π/6)

Answered by bobhans last updated on 24/Sep/20

(2) lim_(x→∞)  e^(x^3  +(√(cos ((1/x^2 )))))  = e^(lim_(x→∞)  x^3 +(√(cos ((1/x^2 )))))    = e^∞  = ∞

Answered by bobhans last updated on 24/Sep/20

(3) lim_(x→0)  csc x. sin (sin x) = lim_(x→0)  ((sin (sin x))/(sin x))  [ let sin x = w ; w→0 ]  lim_(x→0)  ((sin (sin x))/(sin x)) = lim_(w→0)  ((sin (w))/w) = 1

Answered by Dwaipayan Shikari last updated on 24/Sep/20

lim_(x→1) sin^(−1) (((1−(√x))/(1+x)))=sin^(−1) ((1/(1+(√x))))=(π/6)