Example 7

In the special theory of relativity, a body which is moving at constant velocity v, -c < v < c, will have mass

and its length in the direction of motion will be

Here m₀, l₀, and c are positive constants denoting the mass at rest (that is, the mass when v = 0), the length at rest, and the speed of light. Suppose the velocity v is infinitely close to the speed of light c, that is, v = c - ε, ε > 0 infinitesimal. Then

which is the square root of a positive infinitesimal. Thus is a positive infinitesimal. Therefore for v infinitely close to c, m is positive infinite and l is positive infinitesimal. That is, a body moving at velocity infinitely close to (but less than) the speed of light has infinite mass and infinitesimal length in the direction of motion. In the notation of limits this means that

Caution:
This example must be understood in the light of our policy of speaking as if a line in physical space really is like the hyperreal line. Actually, there is no evidence one way or the other on whether a line in space is like the hyperreal line, but the hyperreal line is a useful model for the purpose of applications.