Limits are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach. 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. In this treatment, calculus is a collection of techniques for manipulating certain limits. They capture small-scale behavior, just like infinitesimals, but use the ordinary real number system. Limits describe the value of a function at a certain input in terms of its values at nearby input. In the 19th century, infinitesimals were replaced by limits. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals. This approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the Archimedean property. An infinitesimal number dx could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3. Also one can use a set of well known standard limit formulas along with these theorems. These include laws of algebra of limits, squeeze theorem, LHospitals Rule, Taylor expansions and many more for specialized problems. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". Limits are evaluated using theorems meant to evaluate them. Historically, the first method of doing so was by infinitesimals. Calculus is usually developed by manipulating very small quantities.
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