Delta, Epsilon, and Limits
In informal terms, to say that "a function f(x) has a limit at x = c" means that the closer x gets to c, the closer the value of f(x) gets to the limit. More formally, L is the limit of f(x) as x approaches c if and only if, for any ε (epsilon) greater than zero, there is a value δ (delta) greater than zero such that if x is within δ of c (but x is not equal to c), the value of f(x) is within ε of L.
In this activity, you use Sketchpad to explore the definition visually. For a given function and for given values of c, L, and ε, you'll adjust the value of δ to test the definition.
Open the sketch Del_Eps.gsp in the Limits folder. This sketch shows the graph of a second-degree polynomial function y = ax2 + bx + c.
Your teacher may ask you to change the function by changing the values of the constants a, b, and c. You can do so by adjusting the sliders labeled a, b, and c in the sketch.
Steps (Part 1):
Step 1: Observe how the value of ε (controlled by the red ε slider) determines an interval on the y-axis about L. This interval is represented by a vertical red segment.
Also observe how the value of δ (controlled by the blue δ slider) determines an interval on the x-axis about c. This interval is represented by the width of a vertical blue polygon.
Also observe how the intersection of the left and right edges of this blue polygon with the function determines another polygon (horizontal this time). The height of this second green polygon represents the possible y-values of the function when the x-value is restricted to within δ of c.
Step 2: Double-click the button labeled Case 1. This sets the values of c, L, and ε.
The width of the vertical blue polygon represents all possible x-values within δ of c.
The height of the horizontal green polygon represents all possible y-values generated by these x-values.
The red segment on the y-axis represents an interval containing all y-values within ε of L.
The definition of a limit will be satisfied if you can adjust δ in such a way that all the possible y-values represented by the horizontal green polygon fall within this red interval (that is, they fall within ε of L).
Step 3: Adjust the δ slider until the horizontal green polygon falls just within the red (ε) interval on the y-axis. (The circular indicator changes from red to green when all possible y-values lie within the interval.)
Step 4: Record in a table the values of c, L, e, and δ . To make your table, select these four measurements in order, and then choose Tabulate from the Measure menu.
Step 5: Move the ε slider to set a new smaller value of ε, and then see if you can adjust δ to work with this new value of ε. When you've finished your adjustment, record this new set of results in the same table by double-clicking the table.
Step 6: Double-click the Case 2 action button, and repeat Steps 3, 4, and 5„but in Step 4, instead of making a new table, add your results as an entry to the existing table.
Step 7: Repeat for Cases 3 and 4, adding your results to the table each time.
Q1. Was it possible in each case to set a value of δ that satisfies the definition of a limit? (If not, note the values for c, L, and ε for which it was not possible. Did your effort fail because L does not exist, or because you could not adjust δ correctly? What can you conclude about the function as x approaches c?)
Q2. Do you think it would be possible to set such a value of δ for every possible value of c and ε? If not, what values of c and ε would not work?
Steps (Part 2):
Step 8: Follow the same steps described above for the sketch Del_Eps2.gsp. For each case, adjust the δ slider so that the resulting y-values fall within ε of the limit L, and record your results in a table. Record just one set of results for each case.
Q3. Was it possible in each case to set a value of δ that satisfies the definition of a limit? (If not, note the values for c, L, and ε for which it was not possible. Did your effort fail because L does not exist, or because you could not adjust δ correctly? What can you conclude about the function as x approaches c?)
The sketch you're using assumes that the function is strictly increasing or strictly decreasing over the interval from x - δ to x + δ. How does this limitation affect the first (Del_Eps.gsp) sketch? For what values of x and δ does the sketch become inaccurate?