Question

How do you solve: `lim_(n->oo)(ln(1+e^(2n)))/(ln(1+e^(3n)))` ? Thanks!

Answer 1

`2/3`

Explanation:

`(ln(1+e^(2n)))/(ln(1+e^(3n)))=log(e^(2n)(1+e^(-2n)))/log(e^(3n)(1+e^(-3n)))=(log(e^(2n))+log(1+e^(-2n)))/(log(e^(3n))+log(1+e^(-3n)))`

so

`lim_(n->oo)(ln(1+e^(2n)))/(ln(1+e^(3n)))=lim_(n->oo)(2n+log(1+e^(-2n)))/(3n+log(1+e^(-3n))) = 2/3`