Problem: If we write ∣∣∣x2−4∣∣∣<N for all x such that ∣x−2∣<0.01, the smallest value we can use for N is:
Answer Choices:
A. 0.0301
B. 0.0349
C. 0.0399
D. 0.0401
E. 0.0499
Solution:
Since ∣x−2∣<0.01,∣∣∣x2−4∣∣∣=∣x−2∣∣x+2∣. But ∣x+2∣≦∣x∣+2<2.01+2=4.01, since ∣x−2∣<0.01 implies that x<2.01∴∣∣∣x2−4∣∣∣<(.01)(4.01)=.0401
or
∣x−2∣<0.01 implies 1.99<x<2.01∴3.9599<3.9601<x2<4.0401
∴−0401<x2−4<.0401, that is, ∣∣∣x2−4∣∣∣<.0401