find a number δ such that if 0 < |x − 3| < δ then |f(x) − 2| < 0.5. what does δ equal?

Question

Answer 1

The distance from 3 to 2.6 is 0.4 and the distance from 3 to 3.8 is 0.8. If we let δ be the smaller of the two distances (δ = 0.4) then if 0 < lx - 3l < 0.4 , -0.4 < x-3 < 0.4. So, x is between 2.6 and 3.4. According to your graph, the y values (values of f(x)) that correspond to the x-values between 2.6 and 3.4 are all within 0.5 units of 2. So, lf(x) - 2l < 0.5 when 0 < lx - 3l < ô = 0.4.