# Quotient of infinitesimals indeterminate?

• Rasalhague
In summary, the conversation discusses the concepts of infinitesimals and indeterminate forms in nonstandard analysis. Goldblatt's axioms include the sum, product, and quotient of infinitesimals, while Stroyan raises the question of whether the quotient of two infinitesimals should be considered an indeterminate form. The conversation also touches on the definition of the derivative in terms of infinitesimals and limits. Ultimately, it is concluded that the concept of indeterminate forms in nonstandard analysis is similar to that in elementary calculus, and the specific values of infinitesimals and their quotients depend on the function being considered.
Rasalhague
In Lectures on the hyperreals: an introduction to nonstandard analysis, pp. 50-51, Goldblatt includes among his hyperreal axioms that the sum of two infinitesimals is infinitesimal, that the product of an infinitesimal and an appreciable (i.e. nonzero real) number is infinitesimal, and that the quotient of two infinitesimals is an indeterminate form.

Yet Stroyan, p. 50, writes

$$f[x+\delta x]-f[x]=f'[x] \; \delta x + \varepsilon$$

where $\delta x$ and $\varepsilon$ are infinitesimal. How does this not make

$$\frac{f[x+\delta x]-f[x]}{\delta x}$$

an indeterminate form in non-standard analysis? (By indeterminate form, I understand something not defined, something not ascribed any meaning, such as x/0.) I thought defining the derivative as the standard part of a quotient of infinitesimals was one of the main motivations for the hyperreal number system. Is the derivative to be understood in nonstandard analysis as an approximation to something undefined, just as in standard analysis?

It's difficult to ascribe a meaning to quotient of infinitesimals as limits of the type 0/0 can be 0, finite or infinite.

Rasalhague said:
How does this not make

$$\frac{f[x+\delta x]-f[x]}{\delta x}$$

an indeterminate form in non-standard analysis?
Because division is defined for any pair of numbers, as long as the denominator is nonzero.

The derivative at a standard value x is, as I've seen it defined, given by:
$$f'(x) = \mathop{std} \frac{f(x + \epsilon) - f(x)}{\epsilon}$$
if and only if this expression exists and has the same value for all non-zero infinintessimals $\epsilon$. (And the derivative is extended to non-standard numbers by the transfer principle) (the expression could fail to exist becuase the "standard part" is not defined for infinite numbers)

Of course, the definition in terms of limits gives the same result.

Last edited:
Thanks, Hurkyl. So is Goldblatt's prohibition on dividing an infinitesimal by an infinitesimal not part of "standard" nonstandard analysis, or have I misunderstood what indeterminate means? Or is the derivative being defined as the standard part of an indeterminate form, if that's even possible?

Rasalhague said:
have I misunderstood what indeterminate means?
This, I think. From the information given, one cannot determine whether any of the expressions listed are unlimited, appreciable, or infinitessimal, unlike the forms listed in the previous bullets, so that's the sense I'm sure he means "indeterminate". Of course, there is the close analogy to the concept of "indeterminate form" appearing in elementary calculus; the fact that $\epsilon / \delta$ is indeterminate in the sense I just mentioned implies that 0/0 is an indeterminate form in the sense of elementary calculus.

Okay, so indeterminate in that it's not possible to say in general that any infinitesimal divided by any other infinitesimal is always a particular one of these: finite, infinite, infinitesimal or even undefined (as in the case of 0/0); but it may be possible for a given pair of infinitesimal numbers defined by a particular function, depending on the function (and hence on what those numbers are).

Right. For example, if f is a differentiable function e infinitessimal, and x standard, then
f(x+e) - f(x)​
is also infinitessimal, and the quotient
(f(x+e) - f(x))/e​
is infinitessimally close to f'(x).

## 1. What is the definition of "Quotient of infinitesimals indeterminate"?

The quotient of infinitesimals indeterminate refers to a mathematical expression that involves the division of two quantities, where both quantities are approaching zero or have an infinitely small value.

## 2. How is the quotient of infinitesimals indeterminate different from regular division?

The quotient of infinitesimals indeterminate is different from regular division because in regular division, the divisor must have a non-zero value. However, in the quotient of infinitesimals indeterminate, both the dividend and divisor can approach zero.

## 3. What is the significance of the quotient of infinitesimals indeterminate in calculus?

The quotient of infinitesimals indeterminate is significant in calculus because it allows for the study of limits and the concept of infinitesimals. It also helps in the evaluation of derivatives and integrals of functions.

## 4. How is the quotient of infinitesimals indeterminate handled in mathematical proofs?

In mathematical proofs, the quotient of infinitesimals indeterminate is often handled by using the concept of limits. By taking the limit of the quotient as the values of the infinitesimals approach zero, the indeterminate form can be evaluated and solved.

## 5. Can the quotient of infinitesimals indeterminate be defined in terms of real numbers?

No, the quotient of infinitesimals indeterminate cannot be defined in terms of real numbers because it involves the division of quantities that approach zero. Real numbers cannot have a value of zero, therefore it is not possible to define this quotient using real numbers.

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