Question

What is the limit as x turns to infinity of `x(ln(x+3)-ln(x-3))`?

Answer 1

`lim_(x->oo)x(ln(x+3)-ln(x-3))=1`

Explanation:

`lim_(x->oo)x(ln(x+3)-ln(x-3))`

= `lim_(x->oo)xln((x+3)/(x-3))`

= `lim_(x->oo)xln(1+6/(x-3))`

= `lim_(x->oo)ln(1+6/(x-3))^x`

= `lim_(x->oo)ln[(1+6/(x-3))^((x-3)/6)]^((6x)/(x-3))`

= `lim_(x->oo)ln[(1+6/(x-3))^((x-3)/6)]^(6+18/(x-3))`

= `1^6`

= `1`