Answer
$-\frac{1}{2}$
Work Step by Step
$\lim\limits_{t \to 0}(\frac{1}{t\sqrt {1+t}}-\frac{1}{t})=\lim\limits_{t \to 0}\frac{1}{t\sqrt {1+t}}-\frac{\sqrt {1+t}}{t\sqrt {1+t}}=\lim\limits_{t \to 0}\frac{1-\sqrt {1+t}}{t\sqrt {1+t}}*\frac{1+\sqrt {1+t}}{1+\sqrt {1+t}}=\lim\limits_{t \to 0}\frac{1-(1+t)}{t\sqrt {1+t}+t(1+t)}=\lim\limits_{t \to 0}\frac{-t}{t\sqrt {1+t}+t+t^2}=\lim\limits_{t \to 0}\frac{-1}{\sqrt {1+t}+1+t}=\frac{-1}{\sqrt {1+(0)}+1+(0)}=-\frac{1}{2}$
