Sine Wave Example of the Epsilon-Delta Definition of Limit

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This Demonstration illustrates the - definition for the limit of the sinusoid as tends to . You can vary the amplitude , the frequency , and the value of interest, . Then, for each chosen value for , you can find a valid , i.e., a such that whenever , . When a valid has been found, the vertical sides of the box appear green. For an invalid , the vertical sides of the box appear red.

Contributed by: Geoffrey F. Miller, Daniel C. Cheshire, Nell H. Wackwitz, Joshua B. Fagan  (May 2015)
(Texas State University)
Based on a program by: Joseph F. Kolacinski
Open content licensed under CC BY-NC-SA


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The sine function is very important in mathematics and physics. The sine function can also serve as a great example to demonstrate the - definition of the limit of a function.

Recall that a function has limit at provided that for every , there is a such that whenever . This gives rise to the concept of an box (or window or rectangle).

The vertical sides of the box turn green for an appropriate choice of and are red otherwise. Changing the sine wave's amplitude, frequency, and the value of interest affects the valid choices of . For a valid choice of , the sine wave does not cross over the bottom or the top of the box.



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