If `3x+2y=1` is a tangent to `y=f(x)` at `x=1//2`, then `lim_(xrarr0) (x(x-1))/(f((e^(2x))/(2))-f((e^(-2x))/(2)))`

Options

A. `1//3`
B. `1//2`
C. `1//6`
D. `1//7`

Answer

A

Explanation

Slope of `3x+2y=1" is "(-3)/(2)`
`rArr" "f'((1)/(2))=(-3)/(2)`
`therefore" "underset(xrarr0)(lim)(x(x-1))/(f((e^(2x))/(2))=f((e^(-2x))/(2)))`
`=lim(2x-1)/(e^(2x).f'((e^(2x))/(2))+e^(-2x)f'((e^(-2x))/(2)))`
`=(-1)/(f'((1)/(2))+f'((1)/(2)))=(1)/(3)`