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  2. Infinitesimal - Wikipedia

    en.wikipedia.org/wiki/Infinitesimal

    In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective in mathematics, infinitesimal means infinitely small, smaller than any standard real number. Infinitesimals ...

  3. Indeterminate form - Wikipedia

    en.wikipedia.org/wiki/Indeterminate_form

    Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.

  4. Hyperreal number - Wikipedia

    en.wikipedia.org/wiki/Hyperreal_number

    The intuitive motivation is, for example, to represent an infinitesimal number using a sequence that approaches zero. The inverse of such a sequence would represent an infinite number. As we will see below, the difficulties arise because of the need to define rules for comparing such sequences in a manner that, although inevitably somewhat ...

  5. Limit of a function - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_function

    In non-standard calculus the limit of a function is defined by: = if and only if for all , is infinitesimal whenever x − a is infinitesimal. Here R ∗ {\displaystyle \mathbb {R} ^{*}} are the hyperreal numbers and f* is the natural extension of f to the non-standard real numbers.

  6. Nonstandard calculus - Wikipedia

    en.wikipedia.org/wiki/Nonstandard_calculus

    Example 1: a function f is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.

  7. Nonstandard analysis - Wikipedia

    en.wikipedia.org/wiki/Nonstandard_analysis

    For example, if n is a hyperinteger, i.e. an element of *N − N, then 1/n is an infinitesimal. A hyperreal r is limited (or finite) if and only if its absolute value is dominated by (less than) a standard integer. The limited hyperreals form a subring of *R containing the reals. In this ring, the infinitesimal hyperreals are an ideal.

  8. Increment theorem - Wikipedia

    en.wikipedia.org/wiki/Increment_theorem

    In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then Δ y = f ′ ( x ) Δ x + ε Δ x {\displaystyle \Delta y=f'(x)\,\Delta x+\varepsilon \,\Delta x} for some infinitesimal ε , where Δ y = f ( x + Δ x ) − f ( x ...

  9. Deformation (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Deformation_(mathematics)

    Then, the space on the right hand corner is one example of an infinitesimal deformation: the extra scheme theoretic structure of the nilpotent elements in ⁡ ([]) (which is topologically a point) allows us to organize this infinitesimal data. Since we want to consider all possible expansions, we will let our predeformation functor be defined ...