Q13/047) Find limx→1 1/(2(1−(x)^(1/2)))−1/(3(1−(x)^(1/3))).

Put x = u^3 to get 1/(2(1-u^3) – 1/(3(1-u^2)

= (3(1-u^2) – 2 (1 – u^3))/(6(1-u^2)(1-u^3))

Numerator = 3(1-u)(1+u) – 2(1-u)(1+u+u^3))

= (1-u)( 3+ 3u – 2 -2u – 2u^2)

= (1-u)( 1 + u- 2u^2)

= (1-u)(1-u)(1 + 2u)

= (1-u)^2 (1+ 2u)

Denominator = 6(1-u)^2(1+u)(1+u+u^2)

So ratio = ( 1+ 2u)/(6(1+u)(1+u+u^2))

And at x= 1 or u= 1 we get 3/( 6 * 2 * 3) = 1/12