Chapter 1 - Functions and Limits - 1.6 Calculating Limits Using the Limit Laws - 1.6 Exercises - Page 70: 24

$-\frac{1}{9}$

$\lim\limits_{h \to 0}\frac{(3+h)^{-1}-3^{-1}}{h}=\lim\limits_{h \to 0}\frac{\frac{1}{3+h}-\frac{1}{3}}{h}=\lim\limits_{h \to 0}\frac{\frac{3-(3+h)}{3(3+h)}}{h}=\lim\limits_{h \to 0}\frac{-h}{3h(3+h)}=\lim\limits_{h \to 0}\frac{-1}{3(3+h)}=\frac{-1}{3(3+(0))}=\frac{-1}{9}=-\frac{1}{9}$