If `y=f(x^3),z=g(x^5),f^(prime)(x)=tanx ,a n dg^(prime)(x)=secx ,` then find the value of `lim_(x->0)(((dy)/(dz)))/x`

Answer

`(3)/(5)`

Explanation

We have `y=f(x^(3))`
`therefore" "(dy)/(dx)=f'(x^(3))3x^(2)=3x^(2)tan x^(3)`
Also, `z=g(x^(5))`
`therefore" " (dz)/(dx)=g'(x^(5))5x^(4)=5x^(4)sec x^(5)`
`therefore" "(dy)/(dz)=(dy//dx)/(dz//dx)=(3x^(2) tan x^(3))/(5x^(4)sec x^(5))`
`=(3)/(5x^(2))xx(tan x^(3))/(sec x^(5))`
`"or "underset(xrarr0)lim((dy//dz))/(x)=underset(xrarr0)lim(3 tan x^(3))/(5x^(3) sec x^(5))`
`=(3)/(5)`