Intuitive Definition of Limit

In this applet, we see a function

graphed in the

-plane. You can move the blue point on the

-axis and you can change

, the "radius" of an interval centered about that point. The point has

-value

, and you can see the values of

and

. You can use the pre-loaded examples chosen with the slider or type in your own functions with option 10. We say

exists if all the values of

are "really close" to some number whenever

is "really close" to

Explore

Start by dragging the blue point on the -axis. What is the relationship between the red segment on the -axis and the green segment(s) on the -axis?
What does the slider do? Notice that does not ever take on the value of zero. You can "fine tune" by clicking on the slider button then using the left and right keyboard arrows.
As shrinks to , does the green area always get smaller? Does it ever get larger? Does the green area always shrink down to a single point?
Try the various examples in the applet to get a good feeling for your answers in the previous problem.
Example 5 shows a function that is not defined at . Even though has no value, we can make a good estimate of . In this case, tells us what "should" be. Use zooming to estimate this limit.
In Examples 6 and 7, the function is undefined at . (The function truly is undefined, even though the applet shows . Check this yourself by plugging in for in the function). What is the value of ?
Example 8 is a function that gets "infinitely wiggly" around . What happens if and you shrink ? Try this: make and . What will happen as you move slowly toward ? Make a guess before you do it.

Project idea Let

be a function and define

. Be careful to distinguish between

and

You may have to read the definition of

several times and think carefully about the situation. (This mixture of variables

and

comes up again later when we discuss integrals.)

What is when is continuous at ?
What is when has a removable discontinuity at ?
What is when has a jump discontinuity at ? Does it depend on whether or not is defined?
What is when has an infinite discontinuity at ?
Give an example where the domain of is bigger than.
Give an example where the domain of is smaller than .
Give an example where and have the same domain.
Is always a continuous function?
Is it possible for and to be defined but not equal?

This is a modification of an applet designed by Marc Renault.

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