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[Mathematics: Topics: LIMITS, infinity.]

Two mathematical notions of infinity:

• orbiting the unit circle

• following a sine wave

The Unit Circle...

DEFINITION:

1. Graph of the implicit equation x^2 + y^2 = 1; Not a function - divided into +/- the square root of (1-x^2).
2. Graph of the Polar equation r=1 for Theta (angles) 0 radians to 2*Pi radians,
3. Graph on the complex plane of
   1. (-1)^x, x from 0 to 2;
   2. (I)^x, x from 0 to 4;
   3. (-I)^x, x from 0 to 4;

See these demonstrated...here.

The Sine Wave...

DEFINITION:

1. Sine of Theta, with Theta in radians, is the y coordinate of the point on the unit circle bisecting the terminal ray of the angle Theta in standard position: vertex of (0,0) and initial ray coexistant with the positive x axis.
2. Sine of Theta, with Theta in radians, is, the value of the imaginary part of
   1. (-1)^(Theta/Pi)
   2. (I)^(Theta/2*Pi)
   3. (-I)^(-Theta/2*Pi)

See these added to the unit circle...here.

This is somewhat of a tangent.

LIMITS sound like the bounds on the real. A function is said to be continuous at a point if the limit of the function as it approaches from the left (-) EQUALS the value of the function  at the point (0) EQUALS the limit of the function approaching from the right (+).  The area under a (continuous) curve is found by calculating the limit of the sum of rectangular areas under the curve as the number of rectangles approaches infinity, and the base of each rectangle approaches zero.

Infinity has already crept into the picture through using limits to calculate the sum of an infinite number of areas. Infinity is a topic of study in its own right in mathematics.  Infinite series are encountered and their properties examined. Convergent and Divergent series.

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