## Proof of the Limits of Sine and Cosine at Infinity

**Authors:** Jonathan W. Tooker

We develop a representation of complex numbers separate from the Cartesian and polar representations and define a representing functional for converting between representations. We define the derivative of a function of a complex variable with respect to each representation and then we examine the variation within the definition of the derivative. After studying the transformation law for the variation between representations of complex numbers, we show that the new representation has special properties which allow for a consistent modification to the transformation law for the variation which preserves the definition of the derivative. We refute a common proof that the limits of sine and cosine at infinity cannot exist. Then we use the newly defined modified variation in the definition of the derivative to compute the limits of sine and cosine at infinity.

**Comments:** 47 Pages. Greatly improved in v6

**Download:** **PDF**

### Submission history

[v1] 2018-09-11 22:00:35

[v2] 2018-09-13 09:31:38

[v3] 2018-09-14 12:29:27

[v4] 2018-09-20 05:27:16

[v5] 2018-11-06 23:43:46

[v6] 2019-06-27 11:15:56

**Unique-IP document downloads:** 787 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*