Answer
$\lim\limits_{s\to0}\dfrac{[1/\sqrt{1+s}]-1}{s}=-\dfrac{1}{2}.$
Work Step by Step
$\lim\limits_{s\to0}\dfrac{[1/\sqrt{1+s}]-1}{s}=\lim\limits_{s\to0}\dfrac{1-\sqrt{1+s}}{s\sqrt{1+s}}$
$=\lim\limits_{s\to0}[\dfrac{1-\sqrt{s+1}}{s\sqrt{s+1}}\times\dfrac{1+\sqrt{s+1}}{1+\sqrt{s+1}}]$
$=\lim\limits_{s\to0}\dfrac{(1)^2-(\sqrt{s+1})^2}{s\sqrt{s+1}(1+\sqrt{s+1})}=\lim\limits_{s\to0}\dfrac{-1}{\sqrt{s+1}(1+\sqrt{s+1})}$
$=\dfrac{-1}{\sqrt{1+0}(1+\sqrt{0+1})}=-\dfrac{1}{2}.$
