Why Do We Use Minimum and Maximum Values When Proving Limits of Epsilon?

Why Do We Use Minimum and Maximum Values When Proving Limits of Epsilon?

Why Do We Use Minimum and Maximum Values When Proving Limits of Epsilon?

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.