Question

What is the limit of `ln(2x)-ln(1+x)` as x goes to infinity?

Answer 1

I found `ln(2)`

Explanation:

Let us use a rule of the logs:
`lim_(x->oo)[ln(2x)-ln(1+x)]=`
`lim_(x->oo)[ln((2x)/(1+x))]=` collect `x`:
`lim_(x->oo)[ln((2cancel(x))/(cancel(x)(1/x+1)))]=`
as `x->oo` then `1/x->0`
`=ln(2)`

Answer 2

Use the properties of `ln` to rewrite as a single `ln` then use continuity of `ln`.

Explanation:

Rewrite as `lnu`, tghen use the following.

Because `ln` is continuous on `(0,oo)`, we have

if `lim_(xrarroo)u = L` for some number `L`, then

`lim_(xrarroo) lnu = ln(lim_(xrarroo)u)`.

If you just want the answer it is `ln2`.