Since we're looking at how far x moves from 5, and how far f(x) moves from f(5), we have c = 5. Therefore f(c) = f(5) = 4(5) + 1 = 21. Since we want |f(x) – f(5)| < 0.1, we have ε = 0.1. Now we follow our recipe. - We write down the inequality f(c) – ε < f(x) < f(c) + ε. Now fill in the known variables: 21 – 0.1 < f(x) < 21 + 0.1 so 20.9 < 4x + 1 < 21.1
- Solve the inequality for x.
Subtract 1 from each part of the inequality to find 19.9 < 4x < 20.1, then divide each part by 4 to find 4.975 < x < 5.025.
- Subtract c from all parts of the inequality to find δ.
Since c = 5, we subtract 5 from each part of the inequality to find -0.025 < x – 5 < 0.025 therefore |x – 5| < 0.025. Then δ = 0.025 is the number we want.
We can check our answer by working backwards: If |x – 5| < 0.025, then -0.025 < x – 5 < 0.025 and 4.975 < x < 5.025. Now multiply through by 4 and add 1 to find 19.9 < 4x < 20.1, then 20.9 < 4x + 1 < 21.1 Since f(x) = 4x + 1, we can rewrite this as 20.9 < f(x) < 21.1. Subtract f(5) = 21 from each part to find 20.9 – f(5) < f(x) – f(5) < 21.1 – f(5) -0.1 < f(x) – f(5) < 0.1 Therefore |f(x) - f(5)| < 0.1, which was what we wanted to begin with. |