Let `g(x)={(x^2+xtanx-xtan2x)/(a x+tanx-tan3x),x!=0 0,x=0` If `g^(prime)(0)` exists and is equal to nonzero value `b ,` then `52 b/a` is equal to _____

Explanation

`g'(0)=b=underset(xrarr0)lim(x^(2)+x tan x- xtan 2x)/(x(ax+tan x- tan 3x))`
`underset(xrarr0)lim(x+tan x- tan 2 x)/(ax+tan x- tan 3x)`
`x+(x+(x^(3))/(3)+(2)/(15)x^(5)+...oo)`
`=underset(xrarr0)lim(-(2x+(8x^(3))/(3)+(2)/(15)cdot32x^(5)+...oo))/(ax+(x+(x^(3))/(3)+(2)/(15)x^(5)+...oo))`
`-(3x+(27x^(3))/(3)+(2)/(15)243x^(5)+...oo)`
`=underset(xrarr0)lim(x^(3)(-(7)/(3)+(-62)/(15)x^(2)+...oo))/((a+1-3)x+((1)/(3)-9)x^(3)+(2)/(15)(-242)x^(5)+...oo)`
b can be finite if a+ 1-3=0. Therefore,
`a=2and b=(-(7)/(3))/((1)/(3)-9)=((-7)/(3))((3)/(-26))=(7)/(26)rArr52(b)/(a)=7`