Chapter 9 - Section 9.1 - Sequences - 9.1 Assess Your Understanding - Page 649: 96

See below.

Using a calculator: $a_0=9$ $a_1=\frac{1}{2}(a_0+\frac{89}{a_0})=9.44444$ $a_2=\frac{1}{2}(a_1+\frac{89}{a_1})\approx9.433986$ $a_3=\frac{1}{2}(a_2+\frac{89}{a_2})\approx9.433981$ $a_4=\frac{1}{2}(a_3+\frac{89}{a_3})\approx9.4339811$ $a_5=\frac{1}{2}(a_4+\frac{89}{a_4})\approx9.43398115$ $\sqrt{89}=9.43398113206$, so we got pretty close.