Name: Limit Definition
Objective: To strengthen your understanding of the definition of limit
The previous lab did little more than teach you how to use Mathematica as a graphing calculator. (Take a moment to see that you remember how to plot simple functions — for instance, try plotting y = sqrt(x) in the viewing rectangle [0,5] × [-1,3]. Also, if you did not pick this up on the previous lab, you should take a quick look at the Mathematica Summary” at the end of Lab 2 to see how to define a function f in a notebook. After you have done so, read on.) This lab is different. Here the focus is the Calculus content the lab delivers — it is one of the “common experiences” that would be difficult to deliver the class as a whole any other way. As such, you should read this lab like a textbook — one that requires your participation/interaction — gathering what new insights you can (and asking when you cannot) to the definition of limit from each example presented; don't worry about learning the new commands that appear in this lab.
The first cell you execute in this lab will result in your being asked if you want to evaluate the “Initialization Cell”. It would be easier to say “yes” at this time than to have to evaluate that cell later. The initialization cell itself appears at the very end of the lab (it is easy to spot — it's font looks different). In it are defined two commands that are not part of the standard Mathematica command language — Yinterval[ ] and ShowIntervals[ ]. They've been written for this lab. As you step through the lab, note the difference in syntax for these commands (in particular, they accept ‘f’ instead of the usual ‘f[x]’ as an argument). When a person (such as Professor Brink) programs his own commands, he gets to choose his own syntax. Mathematica is flexible enough to allow programming of commands, utilities, etc. for a wide variety of matematical purposes, and it is this flexibility that is the source of both its power and its sometimes lack of user-friendliness. If you get curious, you should take a peek at the initialization cell.
The section entitled “Given e, Find d ” should be instructive for how to carry out problems like numbers 3-10 in Section 2.4. In the 2nd paragraph of that section you are asked to use the “graphics pointer” to estimate where the horizontal lines intersect the graph (the graph produced by the Yinterval command). The graphics pointer is accessed by highlighting the graph (clicking on it), then holding down the ‘apple’ key. As you move the mouse cursor around on the graph, the graphics pointer functions as a “trace” feature would on a graphing calculator, providing you with the x, y-coordinates where the cursor is positioned.
You are welcome to use Mathematica for the above-mentioned problems of Section 2.4, or to mimick what you are doing here (in the “Given e, Find d” section) on a graphing calculator.
Write a short essay addressing the following question(s): The x-interval that you look for that corresponds to a given value of e is supposed to be “centered at a”. When you find the x-coordinates at the points where the horizontal (blue) lines intersect the graphs, is it always the case that these produce an interval centered at a? If not, then how should you choose d?
This page maintained by:
Thomas L. Scofield
Department of Mathematics and Statistics,
Calvin College