Professor: Erika L.C. King
Email:eking@hws.edu
Office: Lansing 304
Phone: (315) 781-3355
Class Website
Applet and Description by Dr. David Eck
The Precise Definition of the Limit
According to the epsilon/delta definition, $ \small\displaystyle \lim_{x\to a}f(x)=L$ if for each positive number, $ \small\varepsilon$, it is possible to find a positive number, $\small \delta$, such that if $\small x$ is a number with $\small 0<\vert x-a\vert<\delta$, it will also be true that $\small f$ is defined at $\small x$ and $\small \vert f(x)-L\vert<\varepsilon$.
Description of Applet Layout
In the applet shown below, the graph of a function, $\small f(x)$, is shown, and numbers $\small a$ and $\small L$ have been picked for you. You can change the values of epsilon and delta by dragging the sliders at the bottom of the applet or by typing new numbers into the input boxes. On the graph, a green line represents the vertical line $\small x=a$, and a red line represents the horizontal line $\small y=L$. The pink area, together with the yellow area, consists of the points $\small (x,y)$ for which $\small \vert y-L\vert<\varepsilon$. The light green area, together with the yellow area, consists of the points $ \small (x,y)$ for which $\small \vert x-a\vert<\delta$. You can vary the widths of these regions by changing the values of epsilon and delta. The graph of the function is shown in black.
How the Applet Can Be Used to Understand the Precise Definition
What has to be true in this applet if the limit of $\small f(x)$ as $\small x$ approaches $\small a$ is in fact equal to $\small L$? It means that you can set the epsilon slider to any value you like, except zero. Then, no matter what positive epsilon you've chosen, it must be possible to adjust the delta slider so that the graph passes through the yellow region without straying into the green region. (A point $ \small (x,f(x))$ in the green region represents a value of $\small x$ such that $\small \vert x-a\vert<\delta$ but $\small \vert f(x)-L\vert\ge\epsilon$. This is bad.) Note that if epsilon is made smaller, then delta might also have to be made smaller. If the limit is $\small L$, you will always be able to make delta small enough to work. If you encounter an epsilon for which no value of delta will work, then the limit is not equal to $\small L$. (The limit could still exist, if it's equal to some other value than $\small L$.) Now that you have read this, start up the applet and experiment with the first example as noted below.
The Applet
Experimenting with the First Example
In the example that is shown when the applet first starts up, the delta value is too big for the epsilon that is selected. The graph hits the green area. However, a smaller value of delta will work. Change delta to 0.1, for example. To get a better view, you can zoom in on a point by clicking on it. Click on the point $\small (a,L)$ two or three times. (This is the point where the red and green lines intersect.) This will enlarge the graph nicely. Note that you will probably also want to enlarge the size of the window to obtain a bigger picture.
Now, try a smaller value of epsilon. Set epsilon to 0.1. The value delta = 0.1 is now too big. But if you reduce it, say to delta = 0.04, it works again. If you reduce epsilon again, you might be forced to reduce delta, and so on. If this process can be repeated infinitely - no matter how small epsilon is, you can find a delta - then the limit as $\small x$ approaches $\small a$ is $\small L$. If you ever get stuck - if you find a positive epsilon for which no positive delta works - then the limit as $\small x$ approaches $\small a$ is not $\small L$. On a computer, of course, you can't really repeat this process infinitely, since the computer can't deal with arbitrarily small numbers. But it can help you understand what is going on.
Besides the startup example, the applet contains four other examples. To load one of the examples, select it from the pop-up menu at the top of the applet and click the "Load Example" button. For each example, look at what happens for $\small \varepsilon$ = 0.1, 0.01, and 0.001. For each value of $ \small \varepsilon$, see if you can find a value of $\small \delta$ such that for every $\small x$ with $\small 0<\vert x-a\vert<\delta$, it follows that $\small \vert f(x)-L\vert<\varepsilon$. In your journal, report your results (including the numerical value of each $\small \delta$, if you can find one that works). For each example, carefully state what you can conclude about $\small \displaystyle \lim_{x\to a}f(x)$ and why.
Don't forget that you can click at a point to zoom in on it!
Last modified: Thursday 12 September 12:40:07 EDT 2011