Precise definition of infinite limit

We say that [i]the limit of [/i][math]f\left(x\right)[/math][i] is [/i][math]\infty[/math][i] as [/i][math]x[/math][i] approaches [/i][math]c[/math] if, for every number [math]B>0[/math] , there exists a corresponding number [math]\delta>0[/math] such that [math]f\left(x\right)>B[/math] whenever [math]0<\mid x-c\mid<\delta[/math]. In this interactive figure, you can move the point [math]c[/math] on the [math]x[/math]-axis. You can change [math]B[/math] and [math]\delta[/math] independently using the sliders or enter [math]\delta[/math] itself (even as a formula involving [math]B[/math] ). You can also change the function, by using the slider or by entering a function. When proving that [math]\lim_{x\to c}f\left(x\right)=\infty[/math], the goal is to KNOW that for every [math]B[/math], there is always a choice of [math]\delta[/math] such that if [math]0<\mid x-c\mid<\delta[/math], then you know [math]f\left(x\right)>B[/math] must be true. Remember that what happens at [math]x=c[/math] is not important, and that is why [math]0<|x-c|[/math] is part of the definition.[br][br]You can see this graphically if for every [math]B[/math] (which produces the yellow region above [math]B[/math]), you can find [math]\delta[/math] (the blue tolerance band around [math]c[/math]). The goal is to produce a [math]\delta[/math] so that if the function is in the blue band ([math]0<\mid x-c\mid<\delta[/math]), it must also be in the yellow region ([math]f\left(x\right)>B[/math]). If that is possible, regardless of [math]B[/math], the limit is [math]\infty[/math]. If that is not always possible, the limit isn't [math]\infty[/math].[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]c[/math] as specified by the problem under consideration.[br]3. You are now ready to see how [math]B[/math] and [math]\delta[/math] relate. Check the box to show [math]B[/math] and how that gives us a tolerance band of [math]\infty[/math]. You can experiment some to see how changing [math]B[/math] to see how [math]B[/math] changes the tolerance band of [math]\infty[/math].[br]4. You want to see what happens as [math]x[/math] approaches [math]c[/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(x)[/math] changes and that sometimes [math]f(x)[/math] is greater than [math]B[/math] and sometimes [math]f(x)[/math] is not.[br]5. Now check to show [math]\delta[/math]. As you move [math]x[/math], you can see if [math]x[/math] is within the [math]\delta[/math] tolerance band of [math]c[/math] or not. Does moving [math]x[/math] into the [math]\delta[/math] tolerance band of [math]c[/math] force [math]f(x)>B[/math]? If not, is there another choice of [math]\delta[/math] that would accomplish this? You can change [math]\delta[/math] using the slider or by entering a value for [math]\delta[/math]. You can use a formula involving [math]B[/math], but this value does not update if you vary [math]B[/math] later.[br]
[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]

Information: Precise definition of infinite limit