`(lim)_(xvec0^+)1/(xsqrt(x))("a t a n"^(-1)(sqrt(x))/a-btan^(-1)(sqrt(x))/b)` has the value equal to `(a-b)/3` b. 0 c. `((a^2-b^2))/(6a^2b^2)` d.`` `(a^2-b^2)/(3a^2b^2)`

Options

A. `(a-b)/(3)`
B. 0
C. `((a^(2)-b^(2)))/(6a^(2)b^(2))`
D. `(a^(2)-b^(2))/(3a^(2)b^(2))`

Answer

D

Explanation

`underset(xrarr0^(+))(lim)(1)/(xsqrtx)(atan^(-1).(sqrtx)/(a)-b tan^(-1).(sqrtx)/(b))`
`=underset(xrarr0)(lim)((a)/((1+(x)/(a^(2)))).(1)/(a).(1)/(2sqrtx)-(b)/((1+(x)/(b^(2)))).(1)/(b).(1)/(2sqrtx))/((3)/(2).sqrtx)`
`=underset(xrarr0)(lim)[(a^(2))/((a^(2)+x)x)-(b^(2))/((b^(2)+x)x)].(1)/(3)`
`=(a^(2)-b^(2))/(3a^(2)b^(2))`