Question Number 115174 by bemath last updated on 24/Sep/20

    lim_(x→1)  ((tan (cos^(−1) ((1/x))))/( (√(x−1)))) = ?

Answered by bobhans last updated on 24/Sep/20

 let cos^(−1) ((1/x)) = ψ ⇒ (1/x) = cos ψ    and tan ψ = (√(x^2 −1)) , then   lim_(x→1)  ((tan (cos^(−1) ((1/x))))/( (√(x−1)))) = lim_(x→1)  ((√(x^2 −1))/( (√(x−1))))   = lim_(x→1)  (√(x+1)) = (√2)

Answered by Olaf last updated on 24/Sep/20

(1/(cos^2 θ)) = 1+tan^2 θ  tanθ = (√((1/(cos^2 θ))−1)) if tanθ > 0  lim_(x→1) ((√((1/(cos^2 (cos^(−1) (1/x))))−1))/( (√(x−1))))  lim_(x→1) ((√((1/(((1/x))^2 ))−1))/( (√(x−1))))  lim_(x→1) ((√(x^2 −1))/( (√(x−1))))  lim_(x→1) (√( x+1)) = (√2)