Question

What is the limit of `(ln(x+2)-ln(x+1))` as x approaches `oo`?

Answer 1

`ln(x+2)-ln(x+1)=color(green)(0)`

Explanation:

`ln(x+2)-ln(x+1) = ln((x+2)/(x+1))`

`color(white)("XXXXXXXXXXXXX")=ln((1+2/x)/(1+1/x))`

`:. lim_(xrarroo)(ln(x+2)-ln(x+1))`

`color(white)("XXXXX")=lim_(xrarroo)ln((1+2/x)/(1+1/x))`

`color(white)("XXXXX")=ln((1+0)/(1+0))`

`color(white)("XXXXX")=ln(1)`

`color(white)("XXXXX")=0color(white)("XXXX")`since `e^0=1`