Building Understanding of the Epsilon-Delta Definition of a Limit via Graphic Representations (ARC)
Swarup Ghosh, Southwestern Oklahoma State University
Michael Hardy, Southeastern Oklahoma State University
This activity is intended to support construction of meaningfulness for the ε–δ definition of the limit of a function through manipulation, analysis and interpretation of dynamic, graphic representations. Particular emphasis is placed on connecting components of graphs to the ε–δ definition of the limit. Students manipulate a dynamic sketch of a function and propose a value for a proposed limit L as x approaches a value a, if they think one exists. A point on the function of interest can then be altered, as can the values for L and a, by either dragging a point in the graph or a slider. Students then apply the definition of a limit to determine whether or not for a chosen ε, does a given δ value determine a neighborhood about a, such that if a value is within δ units of a, must the corresponding f(x) or y-values be within ε units of L? With that in mind and with the aid of dynamic sketches, students explore whether or not it is possible to find f(x) values that lie outside an open rectangular area bounded by y = L ± ε and x = a ± δ. If not, students can adjust the value of δ to see if such adjustments yield an open neighborhood about a such that the f(x) values for the x-values in that neighborhood of a lie within the open rectangular region, i.e., are within ε units of the proposed limit L. This process nurtures students’ construction of conceptual understanding of the ε–δ definition of the limit of a function and allow them to interpret that definition from a graphic or geometric perspective as well as an algebraic perspective.
Links to Dynamic Sketches
Geometer’s Sketchpad Sketches: Sketches created using The Geometer’s Sketchpad software are accessible via a separate file entitled ε-δ Skpad Sketches.gsp, and that file can be downloaded via the following link:
https://drive.google.com/file/d/1ib11tu8GHE-ErhfykwbMadugeOTSXVZ-/view?usp=sharing
Be aware that there are no Sketchpad sketches for Examples 7 & 8 because the software did not represent those examples well.
This work is licensed under CC BY-NC-SA 4.0
