People use minimum and maximum values when proving limits of epsilon, because epsilon is a range around a limit point. To have a limit, one has to approach that limit from both the negative and positive directions.
In calculus, y= f(x). Limits describe the approach to a number on either side of the y-axis. For example, if f(x) equals x=1, but x does not equal 2, x becomes undefined when one gets to 2. It remains defined right up to the point at which one bumps into 2 on the x-axis. It resumes being defined as soon as one passes 2 on the x-axis. This means one can reach up to 2 without arriving at 2, for example,1.9, and one can reach 2 from the right side of the number line without actually getting there, for example, 2.1.
As one approaches a point on the x-axis, one is also approaching it from the f(x). So wherever one is on the x-axis, one is a corresponding distance away on the y-axis. Epsilon is the range approaching the limit from either a positive or negative direction. Because epsilon is a range of numbers, one has a maximum number and a minimum number.