A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.
Archimedes used the method of exhaustion to calculate the area under a parabola.
Cauchy around 1840. Lithography by Zéphirin Belliard after a painting by Jean Roller.
Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.
Alhazen, 11th-century Arab mathematician and physicist
Cauchy in later life
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
The title page of a textbook by Cauchy.
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Leçons sur le calcul différentiel, 1829
Maria Gaetana Agnesi
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.

Baron Augustin-Louis Cauchy (, ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics.

- Augustin-Louis Cauchy

He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors.

- Augustin-Louis Cauchy

Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis.

- Mathematical analysis

In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

- Calculus

In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler.

- Mathematical analysis

Several mathematicians, including Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere "notions" of infinitely small quantities.

- Calculus
A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications in science and engineering.

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The limit as: x → x0+ ≠ x → x0−. Therefore, the limit as x → x0 does not exist.

Limit of a function

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Not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.

Not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.

The limit as: x → x0+ ≠ x → x0−. Therefore, the limit as x → x0 does not exist.
The function without a limit, at an essential discontinuity
The limit of this function at infinity exists.
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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

In his 1821 book Cours d'analyse, Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of y=f(x) by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y, while claims that he used a rigorous epsilon-delta definition in proofs.