`If underset(trarrx)lim(e^(t)f(x)-e^(x)f(t))/((t-x)(f(x))^(2))=2 andf(0)=(1)/(2),` then find the value of `f'(0).`

Options

A. 4
B. 2
C. 0
D. 1

Explanation

`underset(trarrx)lim(e^(t)f(x)-e^(x)f(t))/((t-x)(f(x))^(2))=2`
`"or "underset(trarrx)lim(e^(t)f(x)-e^(x)f'(t))/(1cdot(f(x))^(2))=2" (Using L' Hospital Rule)"`
`"or "(e^(x)f(x)-e^(x)f'(x))/(1cdot(f(x))^(2))=2`
`"or "(d)/(dx)((e^(x))/(f(x)))=2`
`"or "(e^(x))/(f(x))=2x+c`
given f(0)=2
`"or "(1)/(f(0))=c`
`"or "c=2`
`"or "f(x)=(e^(x))/(2x+2)`
`therefore" "f'(x)=(e^(x)(2x+2)-2e^(x))/((2x+2)^(2))`
Hence, `(f'(0)=0`