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 < 4*x* + 1 < 21.1
- Solve the inequality for
*x*. Subtract 1 from each part of the inequality to find 19.9 < 4*x* < 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 < 4*x* < 20.1, then 20.9 < 4*x* + 1 < 21.1 Since *f*(*x*) = 4*x* + 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. |