Page "Infinitesimal calculus" Paragraph 9
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Standard calculus is based on the approach that took Weierstrass, replacing infinitesimals by limits.
They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system.
Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers.
Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.
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