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Rasalhague

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*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

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

where [itex]\delta x[/itex] and [itex]\varepsilon[/itex] are infinitesimal. How does this not make

[tex]\frac{f[x+\delta x]-f[x]}{\delta x}[/tex]

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?