Delta-Epsilon Limit demonstration
- Mike May
The intuitive definition of limit is that for any y-range around a limit point there is an x-range so the graph exits through the sides of the box rather than though the top or bottom. One practical way to get an idea if a point is a limit is to find values of delta that work when epsilon in .1, .01, and .001. This applet lets you look at a variety of functions. The value c, where we are taking the limit, and L, our candidate for the limit, can be set either by a slider or text box on the left hand panel. The values for and , the x and y range of the window are set in the right window. Formally, we have a limit if for every we can find a so the graph goes out the sides rather than the top or bottom of the box
The function choice slider lets you either consider a preloaded function or one of your own construction. These are the preset functions along with features to examine. 1) A parabola. We have a limit at each point but the width of a good window depends on the value for c as well as the height of the box. 2) A straight line. Once again we have a limit at each point, but the width of a good window does not depend on the value of c. 3) A parabola with a hole at x=1. Removable holes do not have an impact on limits. 4) sin(x)/x. This is a standard limit that will need to get evaluated. It is worthwhile noticing how nice the graph is. 5) sin(1/x)+1. If we focus on c=0, the graph is going up and down so fast that we cannot find a limit. At any other point we can find a limit. 6) x sin(1/x) + 1. This is a modification of the previous example. It still wiggles, but the wiggles get smaller so we can have a limit. 7) This function uses a "greatest integer function" in its construction. It has a lot of breaks. 8) This function goes off to positive infinity when c=0 so depending on our convention either we cannot have a limit, or the limit is positive infinity. 9) This modifies the previous example to go to positive and negative infinity at the same time. 10) User choice function: This allow you to enter your favorite function. It should be noted that the size on delta and epsilon is limited so the box is visible on both views. To use a small delta or epsilon, the values of xmin, xmax, ymin, and ymax may need to be adjusted.