Let f(x) be defined for all x in R such ...

Let f(x) be defined for all `x in R` such that `lim_(xrarr0) [f(x)+log(1-(1)/(e^(f(x))))-log(f(x))]=0`. Then f(0) is

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The correct Answer is:

`underset(xrarr0)(lim)[f(x)+log(1-(1)/(e^(f(x))))-log(f(x))]=0`
`rArr" "underset(xrarr0)(lim)[f(x)+log((e^(f(x))-1)/(e^(f(x))))-log(f(x))]=0`
`rArr" "underset(xrarr0)(lim)[f(x)+log((e^(f(x))-1)/(f(x)))-f(x)]=0`
`rArr" "log(underset(xrarr0)(lim)((e^(f(x))-1)/(f(x))))=0`
`rArr" "underset(xrarr0)(lim)((e^(f(x))-1)/(f(x)))=1`
`rArr" "underset(xrarr0)(lim)f(x)=0`
`rArr" "f(0)=0`

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