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

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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.**

To find the lt{ln(1+nx)/x )= 3 as x--> 0.

Therefore Lt (1/x) ln (1+nx) = 3 as x-->0

Lt ln (1+nx) ^(1/x) = 3 as x--> 0............(1)

LHS : ln (1+n/m) )^m = 3 , where 1/x = m and x = 1/m and as x--> 0, m -->infinity.

ln (1+n/m)^n = ln (e^n) as x--> infinity. Since Lt (1+a/x)^x = e^a as x-->infinity is a standard limit.

Therefore LHS = ln(e^n ) = 3 .

Therefore n = 3