Question

Answer 1

`3)lim (\sqrt{n^2+3n-1}-2n+3)`

`=lim n(\sqrt{1+3/n-1/[n^2]}-2+3/n)=-oo`

`4)lim(\sqrt{n^2+3n-1}-n+3)`

`=lim [n^2+3n-1-n^2+6n-9]/[\sqrt{n^2+3n-1}+n-3]`

`=lim [9n-10]/[\sqrt{n^2+3n-1}+n-3]`

`=lim [9-10/n]/[\sqrt{1+3/n-1/[n^2]}+1-3/n]=9/[1+1]=9/2`

`5)lim n(\sqrt{n^2+1}+3n)`

`=lim n^2(\sqrt{1+1/[n^2]}+3)=+oo`

`6)lim n(\sqrt{n^2+1}-n)`

`=lim n[n^2+1-n^2]/[\sqrt{n^2+1}+n]`

`=lim n/[\sqrt{n^2+1}+n]`

`=lim 1/[\sqrt{1+1/[n^2]}+1]=1/2`