how do you take a limit of [ln(1+e^n)]/2n as n goes to infinity?

Answer 1

Take the limit of the ratio of the derivatives of numerator and denominator, with respect to n.

That ratio is the limit of
e^n/[2(1 + e^n)]

which approaches 1/2 since 1 becomes negligible compared to e^n, as n goes to infinity.