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)