Elementary Calculus: Rules For Infinitesimal, Finite and Infinite Numbers

Rules For Infinitesimal, Finite and Infinite Numbers

Assume that ε, δ are infinitesimals; b, c are hyperreal numbers that are finite but not infinitesimal; and H, K are infinite hyperreal numbers.

(i) Real numbers:

The only infinitesimal real number is 0.

Every real number is finite.

(ii) Negatives:

-ε is infinitesimal.

-b is finite but not infinitesimal.

-H is infinite.

(iii) Reciprocals:

If ε ≠ 0, 1/ε is infinite.

1/b is finite but not infinitesimal.

1/H is infinitesimal.

(iv) Sums:

ε + δ is infinitesimal.

b + ε is finite but not infinitesimal.

b + c is finite (possibly infinitesimal).

H + ε and H + b are infinite.

(v) Products:

5 · ε and b · ε are infinitesimal.

b · c is finite but not infinitesimal.

H · b and H · K are infinite.

(vi) Quotients:

ε/b, ε/H, and b/H are infinitesimal.

b/c is finite but not infinitesimal.

b/ε, H/ε, and H/b are infinite, provided that ε ≠ 0.

(vii) Roots:

If ε >0,is infinitesimal.

If b > 0,is finite but not infinitesimal.

If H >0,.is infinite.

Notice that we have given no rule for the following combinations:

ε/δ, the quotient of two infinitesimals.

H/K, the quotient of two infinite numbers.

Hε, the product of an infinite number and an infinitesimal.

H+K, the sum of two infinite numbers.

Each of these can be either infinitesimal, finite but not infinitesimal, or infinite, depending on what ε, δ, H, and K are. For this reason, they are called indeterminate forms.

Here are three very different quotients of infinitesimals.

is infinitesimal (equal to ε).

is finite but not infinitesimal (equal to 1).

is infinite (equal to )

Table 1.5.1 shows the three possibilities for each indeterminate form.

Table 1.5.1

indeterminate form

Examples

infinitesimal

finite (equal to 1)

finite

ε/δ

ε²/ε

ε/ε

ε/ε²

H/K

H/H²

H/H

H²/H

Hε

H · (1/H²)

H · (1/H)

H² · (1/H)

H + K

H + (-H)

(H + 1) + (-H)

H + H

Here are some examples which show how to use our rules.

Example 1

The next three examples are quotients of infinitesimals.

Example 2

Example 3

Example 4

Example 5