In this applet, we see a function [math]f[/math] graphed in the [math]xy[/math]-plane. [br]You can move the blue point on the [math]x[/math]-axis and you can change [math]\delta[/math], the "radius" of an interval centered about that point. [br]The point has [math]x[/math]-value [math]c[/math], and you can see the values of [math]c[/math] and [math]f(c)[/math]. [br]You can use the pre-loaded examples chosen with the slider or type in your own functions with option 10.[br][br]We say [math]\lim_{x\to c} f(x)[/math] exists if all the values of [math]f(x)[/math] are "really close" to some number whenever [math]x[/math] is "really close" to [math]c[/math].

[b]Explore[/b][br][br][list][br][*]Start by dragging the blue point on the [math]x[/math]-axis. What is the relationship between[br]the red segment on the [math]x[/math]-axis and the green segment(s) on the [math]y[/math]-axis?[br][*]What does the [math]\delta[/math] slider do? Notice that [math]\delta[/math] does not ever take on the[br]value of zero. You can "fine tune" [math]\delta[/math] by clicking on the slider button[br]then using the left and right keyboard arrows.[br][*]As [math]\delta[/math] shrinks to [math]0[/math], does the green area[br]always get smaller? Does it ever get larger? Does the green area always shrink down[br]to a single point?[br][*]Try the various examples in the applet to get a good feeling for your answers[br]in the previous problem.[br][*]Example 5 shows a function that is not defined at [math]x=1[/math]. Even though [math]f(1)[/math] has[br]no value, we can make a good estimate of [math]\lim_{x\to 1} f(x)[/math]. In this case,[br][math]\lim_{x\to 1} f(x)[/math] tells us what [math]f(1)[/math] "should" be. Use zooming to estimate this limit. [br][*]In Examples 6 and 7, the function is undefined at [math]x=2[/math]. (The function truly is undefined,[br]even though the applet shows [math]f(2) = \infty[/math]. Check this yourself by plugging[br]in [math]2[/math] for [math]x[/math] in the function). What is the value of [math]\lim_{x\to 2} f(x)[/math]?[br][*]Example 8 is a function that gets "infinitely wiggly" around [math]x=1[/math].[br]What happens if [math]c=1[/math] and you shrink [math]\delta[/math]? Try this: make [math]c=1[/math] and [math]\delta=0.001[/math].[br]What will happen as you move [math]c[/math] slowly toward [math]1[/math]? Make a guess before you do it.[br][/list][br][br][b]Project idea[/b][br]Let [math]f(x)[/math] be a function and define [math]g(x) = \lim_{t \to x} f(t)[/math].[br]Be careful to distinguish between [math]t[/math] and [math]x[/math] You may have to read the definition[br]of [math]g(x)[/math] several times and think carefully about the situation. (This mixture of[br]variables [math]x[/math] and [math]t[/math] comes up again later when we discuss integrals.)[br][list][br][*]What is [math]g(c)[/math] when [math]f[/math] is continuous at [math]x = c[/math]?[br][*]What is [math]g(c)[/math] when [math]f[/math] has a removable discontinuity at [math]x = c[/math]?[br][*]What is [math]g(c)[/math] when [math]f[/math] has a jump discontinuity at [math]x = c[/math]?[br]Does it depend on whether or not [math]f(c)[/math] is defined?[br][*]What is [math]g(c)[/math] when [math]f[/math] has an infinite discontinuity at [math]x = c[/math]?[br][*]Give an example where the domain of [math]g(x)[/math] is bigger than[math] f(x)[/math].[br][*]Give an example where the domain of [math]g(x)[/math] is smaller than [math]f(x)[/math].[br][*]Give an example where [math]g[/math] and [math]f[/math] have the same domain.[br][*]Is [math]g(x)[/math] always a continuous function?[br][*]Is it possible for [math]g(c)[/math] and [math]f(c)[/math] to be defined but not equal?[br][/list][br][br]This is a modification of an applet designed by Marc Renault.