Question

How do you find the limit of `sqrt(x)*(sqrt(x+8) - sqrt(x-6))` as x approaches `oo`?

Answer 1

`7`

Explanation:

`sqrt(x)*(sqrt(x+8) - sqrt(x-6))(sqrt(x+8) +sqrt(x-6))/(sqrt(x+8) + sqrt(x-6)) =`
`=sqrt(x)(x+8-(x-6))/(sqrt(x+8) + sqrt(x-6)) =(14sqrt(x))/(sqrt(x+8) + sqrt(x-6))=`
`=sqrt(x)/sqrt(x)14/(sqrt(1+8/x)+sqrt(1-6/x)) = 14/(sqrt(1+8/x)+sqrt(1-6/x))`

then

`lim_(x->oo)sqrt(x)*(sqrt(x+8) - sqrt(x-6))=lim_(x->oo)14/(sqrt(1+8/x)+sqrt(1-6/x))=7`