Determine the n value of lim[ln(1+nx)]/x=3 if x-->0

lim [ln (1+nx)]/x=lim
(1/x)*ln(1+nx)

We'll use the power property of the
logarithm:

lim [ln (1+nx)]/x=lim
ln[(1+nx)^(1/x)]

The limit will override the logarithm and
we'll go near the function (1+nx)^(1/x).

ln lim
(1+nx)^(1/x) = ln lim [(1+nx)^(1/nx)]*n

ln lim
(1+nx)^(1/x)=ln e^n

ln lim (1+nx)^(1/x)=n*ln
e

ln lim
(1+nx)^(1/x)=n

But, from hypothesis, lim [ln
(1+nx)]/x=3, so n=3.