Limits via Estimating Slope

Prerequisites: Graphs, slope.

Goal: To get an intuitive understanding of the idea of a limit. To preview the idea of a derivative.

After starting Maple, click File → New → Worksheet Mode to prepare Maple to accept commands. Then copy and paste the following code into your Maple worksheet. The color is only for your own reference; you may want to edit the purple characters, but probably not the red.

restart: with(plots):
Digits := 40: sz := 5:
f := x -> x^2 - x + sin(300*x)/200:
graph1 := plot(f(x),x=-sz..sz, y=-sz..sz,color=blue):
graph2 := plot( 2*x,x=-sz..sz, y=-sz..sz,color=red):
display(graph1,graph2,scaling=constrained);
In this particular code, the variable sz represents the size of square around (0,0) that will be plotted. It is originally set to 5, so the window is centered at the origin and goes from -5 below 0 to 5 above 0 in both the x and y directions.
  1. The command that begins with "graph1" tells the computer to graph, in blue, the function f(x) that we defined earlier. The next command (that begins "graph2") tells the computer to graph, in red, 2*x. Clearly the line y=2*x does not have the same slope as the function f(x) at (0,0). Adjust the slope of the line graphed in red (currently 2) so the two functions apparently have the same slope at (0,0). (The red line should appear to be tangent to the blue curve at the origin.) What is the apparent slope of f(x)?
  2. Zoom in closer to the point (0,0) by setting sz to 1. Note that we only changed the magnification factor. Does the red line still appear to be tangent to the blue curve? If not, what slope makes the red line appear to be tangent to the blue curve?
  3. Zoom in closer to the point (0,0) by setting sz to 0.1. Note that we only changed the magnification factor. Does the red line still appear to be tangent to the blue curve? If not, what slope makes the red line appear to be tangent to the blue curve?
  4. What do you think will happen if we continue to zoom? Why?
  5. How far do we need to "zoom" in order to guarantee that nothing more unexpected will happen in this case? In general?
Last Update: 2007/06/25