Find the limit as x approaches infinity of ##y=ln( 2x )-ln(1+x)##?

First, keep in mind the properties of logarithms.

So, first simplify the function.

Assume ##f(x) = ln(2x)-ln(1+x)##.

Using properties of natural logarithms, the function becomes

##f(x) = ln((2x)/(1+x))##

Then,

##f(x) = ln(2)+ ln(x/(1+x))##.

Finally, when we take the limit as x approaches infinity, we can take the sum of the limits.

##lim_(x->infty) f(x) = lim_(x->infty)[ln(2)]+ lim_(x->infty)[ln(x/(1+x))]##

##lim_(x->infty)[ln(2)]= ln(2) ##

##lim_(x->infty)[ln(x/(1+x))] =ln(lim_(xrarroo)[x/(1+x)])= ln(1) = 0##

##ln(2) + 0 = ln(2)##

Therefore, the limit as x approaches infinity is ln(2).