Determine the n value of limln1nxx3 if x0Solutionlim ln 1nxx

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

Solution

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.