Precise definition of the limit as x approaches infinity
The limit of as approaches is the number 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 but enter as "e" in that formula). You can change the function by using the slider or by entering a function.
When proving that the limit of as is , the goal is to KNOW that for every , there is always a choice of such that must be true whenever .
You can see this graphically if for every used (which produces the yellow tolerance band around , one can find (the blue region extending past ). The goal is to produce a value of so that if is within the blue region (), it must also be in the yellow band (). If that is possible, regardless of , the limit is . If that is not always possible, the limit is not (perhaps the limit attempted is incorrect, or perhaps the limit does not exist). The viewing window adjusts size as is adjusted and as is adjusted. As you select another example (using the slider) the viewing window adjusts back to a default window.
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 to what seems to be an appropriate value to work with.
3. You are now ready to see how and relate. Check the box to show the tolerance band of . You can then experiment some with 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 within the tolerance band of and sometimes is not.
5. Now check to show . As you move , you can see if is greater than or not. Does making force to be in the tolerance band of ? If not, is there a 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 will have to use e (no easy way to use the keyboard to enter here) and this value does not update if you vary later.