1.6, Precise Definitions of the Limit and the Derivative

In Lecture 1.6 we focus on making the limit concept precise and then using that concept to define the derivative.

The formal definition of the limit

The idea of a limit, that as x goes to a, y goes to L, is now interpreted to mean that given an arbitrary small tolerance around L, one can find a small interval around a so that if x is in the interval around a, y is within the small tolerance from L.  The small tolerance on the y-axis is traditionally denoted by the Greek letter epsilon.  The interval on the x-axis is traditionally balanced around a, so that is is (a-delta, a+delta) for some positive number delta.

The formal definitions of the limit appears in the OpenStax section 2.5. In that section we "quantify closeness" by using epsilon for the tolerance (or "closeness") on the y-axis and delta for the "closeness" on the x-axis.  That OpenStax section has a nice applet that let's one experiment with delta and epsilon.

In my Google Drive folder for lecture 1.6 I also have a short video exploring the Epsilon-Delta game at the Desmos webpage.

The formal definition of the limit is a precise but subtle concept. I have brief lecture notes on it in my Google Drive folder, but the Open Stax section 2.5 is probably a better reference.

The formal definition of the derivative

The IRC, or derivative at a point P, is formally defined as the limit of the ARCs as the second point Q approaches the point Q.  This is discussed in the OpenStax section 3.1.  I explored this concept back in the beginning of the semester, in Lecture 1.2.  I have a short video exploring a Math Insight applet in my Google Drive lecture 1.2 folder.

Additional resources

There are some nice applets driven by Geogebra at this Shippensburg University webpage.