If the equation of the normal to the curve `y=f(x) at x=0` is `3x-y+3=0` then the value of
`underset(xrarr0)(lim)(x^(2))/({f(x^(2))-5f(4x^(2))+4f(7x^(2))})` is

Options

A. `(1)/(3)`
B. `-(1)/(3)`
C. `-(1)/(5)`
D. `(1)/(4)`

Answer

B

Explanation

The line `3x-y+3=0` is normal to the curve `y=f(x)` at `x=0`.
` therefore` (Slope of the normal at `x=0`)=3
`rArr -(1)/(f'(0)=3rArr f'(0) =-(1)/(3)`
The normal `3x-y+3=0` passes through `(0,f(0))`,therefore, `f(0)=3`.
Now,
`lim_(xto0) x^(2^(-)){f(x^2)-5f(4x^2)+4f(7x^2)}^-1`
`=lim_(xto0) (x^2)/(f(x^2)-5f(4x^2)+4f(7x^2))`