We say that the limit of is as approaches if, for every number , there exists a corresponding number such that whenever . In this interactive figure, you can move the point on the -axis. You can change and independently using the sliders or enter itself (even as a formula involving ). You can also change the function, by using the slider or by entering a function. When proving that , the goal is to KNOW that for every , there is always a choice of such that if , then you know must be true. Remember that what happens at is not important, and that is why is part of the definition.
You can see this graphically if for every (which produces the yellow region above ), you can find (the blue tolerance band around ). The goal is to produce a so that if the function is in the blue band (), it must also be in the yellow region (). If that is possible, regardless of , the limit is . If that is not always possible, the limit isn't .
You can do a bit of zooming by holding the shift key and mouse button while on an axis and dragging to change the scale on that axis. Using the scroll wheel on a mouse will also allow for zooming and you can drag the graph to keep the point of interest near the center of the graph. After zooming, it is probably easiest to reload the interactive figure to start another investigation.
This interactive figure is quite involved.
1. You first enter the function or use the slider to select from given examples.
2. You then adjust as specified by the problem under consideration.
3. You are now ready to see how and relate. Check the box to show and how that gives us a tolerance band of . You can experiment some to see how changing to see how changes the tolerance band of .
4. You want to see what happens as approaches , so you now need to check the box to show . You can move to see how changes and that sometimes is greater than and sometimes is not.
5. Now check to show . As you move , you can see if is within the tolerance band of or not. Does moving into the tolerance band of force ? If not, is there another choice of that would accomplish this? You can change using the slider or by entering a value for . You can use a formula involving , but this value does not update if you vary later.

Developed for use with Thomas' Calculus, published by Pearson.