Editing Calculus (section)

=== Limits and infinitesimals ===
{{Main|Limit of a function|Infinitesimal}}
Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by [[infinitesimal]]s. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small".  For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positive [[real number]]. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols <math>dx</math> and <math>dy</math> were taken to be infinitesimal, and the derivative <math>dy/dx</math> was their ratio.<ref name="Bell-SEP" />

The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by the [[epsilon, delta]] approach to [[Limit of a function|limits]]. Limits describe the behavior of a [[function (mathematics)|function]] at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of the [[real number|real number system]] (as a [[metric space]] with the [[least-upper-bound property]]). In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by sequences of smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the limiting behavior for these sequences. Limits were thought to provide a more rigorous foundation for calculus, and for this reason, they became the standard approach during the 20th century. However, the infinitesimal 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.<ref name="Bell-SEP"/>