The limit of [math]f\left(x\right)[/math] as [math]x[/math] approaches [math]\infty[/math] is the number [math]L[/math] if, for every number [math]\epsilon>0[/math] , there exists a corresponding number [math]M>0[/math] such that [math]\mid f\left(x\right)-L\mid<\epsilon[/math] whenever [math]x>M[/math]. [br][br]In this interactive figure, you can move the point [math]L[/math] on the [math]y[/math]-axis. You can change [math]\epsilon[/math] and [i][math]M[/math][/i] independently using the sliders or enter [math]M[/math] itself (even as a formula involving [math]\epsilon[/math] but enter [math]\epsilon[/math] as "e" in that formula). You can change the function by using the slider or by entering a function.[br][br]When proving that the limit of [math]f(x)[/math] as [math]x\to\infty[/math] is [math]L[/math], the goal is to KNOW that for every [math]\epsilon[/math], there is always a choice of [math]M[/math] such that [math]\mid f\left(x\right)-L\mid<\epsilon[/math] must be true whenever [math]x>M[/math].[br][br]You can see this graphically if for every [math]\epsilon>0[/math] used (which produces the yellow tolerance band around [math]L[/math], one can find [math]M[/math] (the blue region extending past [math]M[/math]). The goal is to produce a value of [math]M[/math] so that if [math]x[/math] is within the blue region ([math]x>M[/math]), it must also be in the yellow band ([math]\mid f\left(x\right)-L\mid<\epsilon[/math]). If that is possible, regardless of [math]\epsilon[/math], the limit is [math]L[/math]. If that is not always possible, the limit is not [math]L[/math] (perhaps the limit attempted is incorrect, or perhaps the limit does not exist). The viewing window adjusts size as [math]\epsilon[/math] is adjusted and as [math]M[/math] is adjusted. As you select another example (using the slider) the viewing window adjusts back to a default window.[br][br]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.[br][br]This interactive figure is quite involved.[br][br]1. You first enter the function or use the slider to select from given examples.[br]2. You then adjust [math]L[/math] to what seems to be an appropriate value to work with.[br]3. You are now ready to see how [math]\epsilon[/math] and [math]M[/math] relate. Check the box to show the [math]\epsilon[/math] tolerance band of [math]L[/math]. You can then experiment some with [math]\epsilon[/math] to see how [math]\epsilon[/math] changes the tolerance band of [math]L[/math].[br]4. You want to see what happens as [math]x[/math] approaches [math]\infty[/math], so you now need to check the box to show [math]x[/math]. You can move [math]x[/math] to see how [math]f\left(x\right)[/math] changes and that sometimes [math]f\left(x\right)[/math] is within the [math]\epsilon[/math] tolerance band of [math]L[/math] and sometimes [math]f\left(x\right)[/math] is not.[br]5. Now check to show [math]M[/math]. As you move [math]x[/math], you can see if [math]x[/math] is greater than [math]M[/math] or not. Does making [math]x>M[/math] force [math]f\left(x\right)[/math] to be in the [math]\epsilon[/math] tolerance band of [math]L[/math]? If not, is there a choice of [math]M[/math] that would accomplish this? You can change [math]M[/math] using the slider or by entering a value for [math]M[/math]. You can use a formula involving [math]\epsilon[/math], but will have to use e (no easy way to use the keyboard to enter [math]\epsilon[/math] here) and this value does not update if you vary [math]\epsilon[/math] later.