8.
$\lim(\sqrt{n+1}-\sqrt{n})$
$=\lim\dfrac{1}{\sqrt{n+1}+\sqrt{n}}$
$=\lim\dfrac{ \dfrac{1}{\sqrt{n}} }{\sqrt{1+\dfrac{1}{n}}+1}$
$=0$
9.
$\lim\sqrt{n}(\sqrt{n+1}-\sqrt{n-1})$
$=\lim\sqrt{n}.\dfrac{n+1-n+1}{\sqrt{n+1}+\sqrt{n-1}}$
$=\lim\dfrac{2\sqrt{n}}{\sqrt{n+1}+\sqrt{n-1}}$
$=\lim\dfrac{2}{\sqrt{1+\dfrac{1}{n}}+\sqrt{1-\dfrac{1}{n}} }$
$=\dfrac{2}{1+1}=1$
