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Standard calculus is based on the approach that took Weierstrass, replacing infinitesimals by limits.
Limits describe the value of a function at a certain input in terms of its values at nearby input.
They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system.
In this treatment, calculus is a collection of techniques for manipulating certain limits.
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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