If `lim_(xto0) [1+x+(f(x))/(x)]^(1//x)=e^(3)`, then the value of `ln(lim_(xto0) [1+(f(x))/(x)]^(1//x))` is _________.

Answer

`(2)`

Explanation

`underset(xto0)lim[1+x+(f(x))/(x)]^(1//x)=e^(3)`
or `underset(xto0)lime^(underset(xto0)lim[1+x+(f(x))/(x)-1](1)/(x))=e^(3)`
or `underset(xto0)lime^(underset(xto0)lim[1+(f(x))/(x^(2))])=e^(3)" "" or "underset(xto0)lim(f(x))/(x^(2))=2`
Now, `underset(xto0)lim[1+(f(x))/(x)]^(1/x)=e^(underset(xto0)lim[1+(f(x))/(x)-1](1)/(x))=e^(underset(xto0)lim(f(x))/(x^(2)))=e^(2)`