Solve: Lim x->1 (sqrt(x)-sqrt(2-x^2))/(2x-sqrt(2+2x^2) ?

Solve:

Lim x->1 (sqrt(x)-sqrt(2-x^2))/(2x-sqrt(2+2x^2)

1 Answer

3/2

Explanation:

lim_(x->1)(sqrt(x)-sqrt(2-x^2))/(2x-sqrt(2+2x^2)

(sqrt[x] - sqrt[2 - x^2])/(2 x - sqrt[2 + 2 x^2]) *(sqrt[x] + sqrt[ 2 - x^2])/(2 x + sqrt[2 + 2 x^2])

= (x-2+x^2)/(4x^2-2-2x^2)=((x+2)(x-1))/(2(x+1)(x-1))

= 1/2((x+2)/(x+1)) and then

(sqrt(x)-sqrt(2-x^2))/(2x-sqrt(2+2x^2)) = 1/2((x+2)/(x+1))(2 x + sqrt[2 + 2 x^2])/(sqrt[x] + sqrt[ 2 - x^2])

and then

lim_(x->1)(sqrt(x)-sqrt(2-x^2))/(2x-sqrt(2+2x^2)) = lim_(x->1) 1/2((x+2)/(x+1))(2 x + sqrt[2 + 2 x^2])/(sqrt[x] + sqrt[ 2 - x^2])

= 1/2*3/2*4/2=3/2